Divide $$$5 x^{9} - 4 x^{2} + 2$$$ by $$$5 x + 10$$$

Write the problem in the special format (missed terms are written with zero coefficients):

$$$\begin{array}{r|r}\hline\\5 x+10&5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2\end{array}$$$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{5 x^{9}}{5 x} = x^{8}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$x^{8} \left(5 x+10\right) = 5 x^{9}+10 x^{8}$$$.

Subtract the dividend from the obtained result: $$$\left(5 x^{9}- 4 x^{2}+2\right) - \left(5 x^{9}+10 x^{8}\right) = - 10 x^{8}- 4 x^{2}+2$$$.

$$\begin{array}{r|rrrrrrrrrr:c}&{\color{DeepPink}x^{8}}&&&&&&&&&&\\\hline\\{\color{Magenta}5 x}+10&{\color{DeepPink}5 x^{9}}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{DeepPink}5 x^{9}}}{{\color{Magenta}5 x}} = {\color{DeepPink}x^{8}}\\&-\phantom{5 x^{9}}&&&&&&&&&&\\&5 x^{9}&+10 x^{8}&&&&&&&&&{\color{DeepPink}x^{8}} \left(5 x+10\right) = 5 x^{9}+10 x^{8}\\\hline\\&&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\end{array}$$

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- 10 x^{8}}{5 x} = - 2 x^{7}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$- 2 x^{7} \left(5 x+10\right) = - 10 x^{8}- 20 x^{7}$$$.

Subtract the remainder from the obtained result: $$$\left(- 10 x^{8}- 4 x^{2}+2\right) - \left(- 10 x^{8}- 20 x^{7}\right) = 20 x^{7}- 4 x^{2}+2$$$.

$$\begin{array}{r|rrrrrrrrrr:c}&x^{8}&{\color{Violet}- 2 x^{7}}&&&&&&&&&\\\hline\\{\color{Magenta}5 x}+10&5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&-\phantom{5 x^{9}}&&&&&&&&&&\\&5 x^{9}&+10 x^{8}&&&&&&&&&\\\hline\\&&{\color{Violet}- 10 x^{8}}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{Violet}- 10 x^{8}}}{{\color{Magenta}5 x}} = {\color{Violet}- 2 x^{7}}\\&&-\phantom{- 10 x^{8}}&&&&&&&&&\\&&- 10 x^{8}&- 20 x^{7}&&&&&&&&{\color{Violet}- 2 x^{7}} \left(5 x+10\right) = - 10 x^{8}- 20 x^{7}\\\hline\\&&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\end{array}$$

Step 3

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{20 x^{7}}{5 x} = 4 x^{6}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$4 x^{6} \left(5 x+10\right) = 20 x^{7}+40 x^{6}$$$.

Subtract the remainder from the obtained result: $$$\left(20 x^{7}- 4 x^{2}+2\right) - \left(20 x^{7}+40 x^{6}\right) = - 40 x^{6}- 4 x^{2}+2$$$.

$$\begin{array}{r|rrrrrrrrrr:c}&x^{8}&- 2 x^{7}&{\color{DarkCyan}+4 x^{6}}&&&&&&&&\\\hline\\{\color{Magenta}5 x}+10&5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&-\phantom{5 x^{9}}&&&&&&&&&&\\&5 x^{9}&+10 x^{8}&&&&&&&&&\\\hline\\&&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&-\phantom{- 10 x^{8}}&&&&&&&&&\\&&- 10 x^{8}&- 20 x^{7}&&&&&&&&\\\hline\\&&&{\color{DarkCyan}20 x^{7}}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{DarkCyan}20 x^{7}}}{{\color{Magenta}5 x}} = {\color{DarkCyan}4 x^{6}}\\&&&-\phantom{20 x^{7}}&&&&&&&&\\&&&20 x^{7}&+40 x^{6}&&&&&&&{\color{DarkCyan}4 x^{6}} \left(5 x+10\right) = 20 x^{7}+40 x^{6}\\\hline\\&&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\end{array}$$

Step 4

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- 40 x^{6}}{5 x} = - 8 x^{5}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$- 8 x^{5} \left(5 x+10\right) = - 40 x^{6}- 80 x^{5}$$$.

Subtract the remainder from the obtained result: $$$\left(- 40 x^{6}- 4 x^{2}+2\right) - \left(- 40 x^{6}- 80 x^{5}\right) = 80 x^{5}- 4 x^{2}+2$$$.

$$\begin{array}{r|rrrrrrrrrr:c}&x^{8}&- 2 x^{7}&+4 x^{6}&{\color{Brown}- 8 x^{5}}&&&&&&&\\\hline\\{\color{Magenta}5 x}+10&5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&-\phantom{5 x^{9}}&&&&&&&&&&\\&5 x^{9}&+10 x^{8}&&&&&&&&&\\\hline\\&&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&-\phantom{- 10 x^{8}}&&&&&&&&&\\&&- 10 x^{8}&- 20 x^{7}&&&&&&&&\\\hline\\&&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&-\phantom{20 x^{7}}&&&&&&&&\\&&&20 x^{7}&+40 x^{6}&&&&&&&\\\hline\\&&&&{\color{Brown}- 40 x^{6}}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{Brown}- 40 x^{6}}}{{\color{Magenta}5 x}} = {\color{Brown}- 8 x^{5}}\\&&&&-\phantom{- 40 x^{6}}&&&&&&&\\&&&&- 40 x^{6}&- 80 x^{5}&&&&&&{\color{Brown}- 8 x^{5}} \left(5 x+10\right) = - 40 x^{6}- 80 x^{5}\\\hline\\&&&&&80 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\end{array}$$

Step 5

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{80 x^{5}}{5 x} = 16 x^{4}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$16 x^{4} \left(5 x+10\right) = 80 x^{5}+160 x^{4}$$$.

Subtract the remainder from the obtained result: $$$\left(80 x^{5}- 4 x^{2}+2\right) - \left(80 x^{5}+160 x^{4}\right) = - 160 x^{4}- 4 x^{2}+2$$$.

$$\begin{array}{r|rrrrrrrrrr:c}&x^{8}&- 2 x^{7}&+4 x^{6}&- 8 x^{5}&{\color{Chartreuse}+16 x^{4}}&&&&&&\\\hline\\{\color{Magenta}5 x}+10&5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&-\phantom{5 x^{9}}&&&&&&&&&&\\&5 x^{9}&+10 x^{8}&&&&&&&&&\\\hline\\&&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&-\phantom{- 10 x^{8}}&&&&&&&&&\\&&- 10 x^{8}&- 20 x^{7}&&&&&&&&\\\hline\\&&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&-\phantom{20 x^{7}}&&&&&&&&\\&&&20 x^{7}&+40 x^{6}&&&&&&&\\\hline\\&&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&-\phantom{- 40 x^{6}}&&&&&&&\\&&&&- 40 x^{6}&- 80 x^{5}&&&&&&\\\hline\\&&&&&{\color{Chartreuse}80 x^{5}}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{Chartreuse}80 x^{5}}}{{\color{Magenta}5 x}} = {\color{Chartreuse}16 x^{4}}\\&&&&&-\phantom{80 x^{5}}&&&&&&\\&&&&&80 x^{5}&+160 x^{4}&&&&&{\color{Chartreuse}16 x^{4}} \left(5 x+10\right) = 80 x^{5}+160 x^{4}\\\hline\\&&&&&&- 160 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\end{array}$$

Step 6

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- 160 x^{4}}{5 x} = - 32 x^{3}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$- 32 x^{3} \left(5 x+10\right) = - 160 x^{4}- 320 x^{3}$$$.

Subtract the remainder from the obtained result: $$$\left(- 160 x^{4}- 4 x^{2}+2\right) - \left(- 160 x^{4}- 320 x^{3}\right) = 320 x^{3}- 4 x^{2}+2$$$.

$$\begin{array}{r|rrrrrrrrrr:c}&x^{8}&- 2 x^{7}&+4 x^{6}&- 8 x^{5}&+16 x^{4}&{\color{Green}- 32 x^{3}}&&&&&\\\hline\\{\color{Magenta}5 x}+10&5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&-\phantom{5 x^{9}}&&&&&&&&&&\\&5 x^{9}&+10 x^{8}&&&&&&&&&\\\hline\\&&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&-\phantom{- 10 x^{8}}&&&&&&&&&\\&&- 10 x^{8}&- 20 x^{7}&&&&&&&&\\\hline\\&&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&-\phantom{20 x^{7}}&&&&&&&&\\&&&20 x^{7}&+40 x^{6}&&&&&&&\\\hline\\&&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&-\phantom{- 40 x^{6}}&&&&&&&\\&&&&- 40 x^{6}&- 80 x^{5}&&&&&&\\\hline\\&&&&&80 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&&-\phantom{80 x^{5}}&&&&&&\\&&&&&80 x^{5}&+160 x^{4}&&&&&\\\hline\\&&&&&&{\color{Green}- 160 x^{4}}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{Green}- 160 x^{4}}}{{\color{Magenta}5 x}} = {\color{Green}- 32 x^{3}}\\&&&&&&-\phantom{- 160 x^{4}}&&&&&\\&&&&&&- 160 x^{4}&- 320 x^{3}&&&&{\color{Green}- 32 x^{3}} \left(5 x+10\right) = - 160 x^{4}- 320 x^{3}\\\hline\\&&&&&&&320 x^{3}&- 4 x^{2}&+0 x&+2&\end{array}$$

Step 7

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{320 x^{3}}{5 x} = 64 x^{2}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$64 x^{2} \left(5 x+10\right) = 320 x^{3}+640 x^{2}$$$.

Subtract the remainder from the obtained result: $$$\left(320 x^{3}- 4 x^{2}+2\right) - \left(320 x^{3}+640 x^{2}\right) = - 644 x^{2}+2$$$.

$$\begin{array}{r|rrrrrrrrrr:c}&x^{8}&- 2 x^{7}&+4 x^{6}&- 8 x^{5}&+16 x^{4}&- 32 x^{3}&{\color{Blue}+64 x^{2}}&&&&\\\hline\\{\color{Magenta}5 x}+10&5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&-\phantom{5 x^{9}}&&&&&&&&&&\\&5 x^{9}&+10 x^{8}&&&&&&&&&\\\hline\\&&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&-\phantom{- 10 x^{8}}&&&&&&&&&\\&&- 10 x^{8}&- 20 x^{7}&&&&&&&&\\\hline\\&&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&-\phantom{20 x^{7}}&&&&&&&&\\&&&20 x^{7}&+40 x^{6}&&&&&&&\\\hline\\&&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&-\phantom{- 40 x^{6}}&&&&&&&\\&&&&- 40 x^{6}&- 80 x^{5}&&&&&&\\\hline\\&&&&&80 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&&-\phantom{80 x^{5}}&&&&&&\\&&&&&80 x^{5}&+160 x^{4}&&&&&\\\hline\\&&&&&&- 160 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&&&-\phantom{- 160 x^{4}}&&&&&\\&&&&&&- 160 x^{4}&- 320 x^{3}&&&&\\\hline\\&&&&&&&{\color{Blue}320 x^{3}}&- 4 x^{2}&+0 x&+2&\frac{{\color{Blue}320 x^{3}}}{{\color{Magenta}5 x}} = {\color{Blue}64 x^{2}}\\&&&&&&&-\phantom{320 x^{3}}&&&&\\&&&&&&&320 x^{3}&+640 x^{2}&&&{\color{Blue}64 x^{2}} \left(5 x+10\right) = 320 x^{3}+640 x^{2}\\\hline\\&&&&&&&&- 644 x^{2}&+0 x&+2&\end{array}$$

Step 8

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- 644 x^{2}}{5 x} = - \frac{644 x}{5}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$- \frac{644 x}{5} \left(5 x+10\right) = - 644 x^{2}- 1288 x$$$.

Subtract the remainder from the obtained result: $$$\left(- 644 x^{2}+2\right) - \left(- 644 x^{2}- 1288 x\right) = 1288 x+2$$$.

$$\begin{array}{r|rrrrrrrrrr:c}&x^{8}&- 2 x^{7}&+4 x^{6}&- 8 x^{5}&+16 x^{4}&- 32 x^{3}&+64 x^{2}&{\color{BlueViolet}- \frac{644 x}{5}}&&&\\\hline\\{\color{Magenta}5 x}+10&5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&-\phantom{5 x^{9}}&&&&&&&&&&\\&5 x^{9}&+10 x^{8}&&&&&&&&&\\\hline\\&&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&-\phantom{- 10 x^{8}}&&&&&&&&&\\&&- 10 x^{8}&- 20 x^{7}&&&&&&&&\\\hline\\&&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&-\phantom{20 x^{7}}&&&&&&&&\\&&&20 x^{7}&+40 x^{6}&&&&&&&\\\hline\\&&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&-\phantom{- 40 x^{6}}&&&&&&&\\&&&&- 40 x^{6}&- 80 x^{5}&&&&&&\\\hline\\&&&&&80 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&&-\phantom{80 x^{5}}&&&&&&\\&&&&&80 x^{5}&+160 x^{4}&&&&&\\\hline\\&&&&&&- 160 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&&&-\phantom{- 160 x^{4}}&&&&&\\&&&&&&- 160 x^{4}&- 320 x^{3}&&&&\\\hline\\&&&&&&&320 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&&&&-\phantom{320 x^{3}}&&&&\\&&&&&&&320 x^{3}&+640 x^{2}&&&\\\hline\\&&&&&&&&{\color{BlueViolet}- 644 x^{2}}&+0 x&+2&\frac{{\color{BlueViolet}- 644 x^{2}}}{{\color{Magenta}5 x}} = {\color{BlueViolet}- \frac{644 x}{5}}\\&&&&&&&&-\phantom{- 644 x^{2}}&&&\\&&&&&&&&- 644 x^{2}&- 1288 x&&{\color{BlueViolet}- \frac{644 x}{5}} \left(5 x+10\right) = - 644 x^{2}- 1288 x\\\hline\\&&&&&&&&&1288 x&+2&\end{array}$$

Step 9

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{1288 x}{5 x} = \frac{1288}{5}$$$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $$$\frac{1288 \left(5 x+10\right)}{5} = 1288 x+2576$$$.

Subtract the remainder from the obtained result: $$$\left(1288 x+2\right) - \left(1288 x+2576\right) = -2574$$$.

$$\begin{array}{r|rrrrrrrrrr:c}&x^{8}&- 2 x^{7}&+4 x^{6}&- 8 x^{5}&+16 x^{4}&- 32 x^{3}&+64 x^{2}&- \frac{644 x}{5}&{\color{Purple}+\frac{1288}{5}}&&\\\hline\\{\color{Magenta}5 x}+10&5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&-\phantom{5 x^{9}}&&&&&&&&&&\\&5 x^{9}&+10 x^{8}&&&&&&&&&\\\hline\\&&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&-\phantom{- 10 x^{8}}&&&&&&&&&\\&&- 10 x^{8}&- 20 x^{7}&&&&&&&&\\\hline\\&&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&-\phantom{20 x^{7}}&&&&&&&&\\&&&20 x^{7}&+40 x^{6}&&&&&&&\\\hline\\&&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&-\phantom{- 40 x^{6}}&&&&&&&\\&&&&- 40 x^{6}&- 80 x^{5}&&&&&&\\\hline\\&&&&&80 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&&-\phantom{80 x^{5}}&&&&&&\\&&&&&80 x^{5}&+160 x^{4}&&&&&\\\hline\\&&&&&&- 160 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&&&-\phantom{- 160 x^{4}}&&&&&\\&&&&&&- 160 x^{4}&- 320 x^{3}&&&&\\\hline\\&&&&&&&320 x^{3}&- 4 x^{2}&+0 x&+2&\\&&&&&&&-\phantom{320 x^{3}}&&&&\\&&&&&&&320 x^{3}&+640 x^{2}&&&\\\hline\\&&&&&&&&- 644 x^{2}&+0 x&+2&\\&&&&&&&&-\phantom{- 644 x^{2}}&&&\\&&&&&&&&- 644 x^{2}&- 1288 x&&\\\hline\\&&&&&&&&&{\color{Purple}1288 x}&+2&\frac{{\color{Purple}1288 x}}{{\color{Magenta}5 x}} = {\color{Purple}\frac{1288}{5}}\\&&&&&&&&&-\phantom{1288 x}&&\\&&&&&&&&&1288 x&+2576&{\color{Purple}\frac{1288}{5}} \left(5 x+10\right) = 1288 x+2576\\\hline\\&&&&&&&&&&-2574&\end{array}$$

Since the degree of the remainder is less than the degree of the divisor, we are done.

The resulting table is shown once more:

$$\begin{array}{r|rrrrrrrrrr:c}&{\color{DeepPink}x^{8}}&{\color{Violet}- 2 x^{7}}&{\color{DarkCyan}+4 x^{6}}&{\color{Brown}- 8 x^{5}}&{\color{Chartreuse}+16 x^{4}}&{\color{Green}- 32 x^{3}}&{\color{Blue}+64 x^{2}}&{\color{BlueViolet}- \frac{644 x}{5}}&{\color{Purple}+\frac{1288}{5}}&&\text{Hints}\\\hline\\{\color{Magenta}5 x}+10&{\color{DeepPink}5 x^{9}}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{DeepPink}5 x^{9}}}{{\color{Magenta}5 x}} = {\color{DeepPink}x^{8}}\\&-\phantom{5 x^{9}}&&&&&&&&&&\\&5 x^{9}&+10 x^{8}&&&&&&&&&{\color{DeepPink}x^{8}} \left(5 x+10\right) = 5 x^{9}+10 x^{8}\\\hline\\&&{\color{Violet}- 10 x^{8}}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{Violet}- 10 x^{8}}}{{\color{Magenta}5 x}} = {\color{Violet}- 2 x^{7}}\\&&-\phantom{- 10 x^{8}}&&&&&&&&&\\&&- 10 x^{8}&- 20 x^{7}&&&&&&&&{\color{Violet}- 2 x^{7}} \left(5 x+10\right) = - 10 x^{8}- 20 x^{7}\\\hline\\&&&{\color{DarkCyan}20 x^{7}}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{DarkCyan}20 x^{7}}}{{\color{Magenta}5 x}} = {\color{DarkCyan}4 x^{6}}\\&&&-\phantom{20 x^{7}}&&&&&&&&\\&&&20 x^{7}&+40 x^{6}&&&&&&&{\color{DarkCyan}4 x^{6}} \left(5 x+10\right) = 20 x^{7}+40 x^{6}\\\hline\\&&&&{\color{Brown}- 40 x^{6}}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{Brown}- 40 x^{6}}}{{\color{Magenta}5 x}} = {\color{Brown}- 8 x^{5}}\\&&&&-\phantom{- 40 x^{6}}&&&&&&&\\&&&&- 40 x^{6}&- 80 x^{5}&&&&&&{\color{Brown}- 8 x^{5}} \left(5 x+10\right) = - 40 x^{6}- 80 x^{5}\\\hline\\&&&&&{\color{Chartreuse}80 x^{5}}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{Chartreuse}80 x^{5}}}{{\color{Magenta}5 x}} = {\color{Chartreuse}16 x^{4}}\\&&&&&-\phantom{80 x^{5}}&&&&&&\\&&&&&80 x^{5}&+160 x^{4}&&&&&{\color{Chartreuse}16 x^{4}} \left(5 x+10\right) = 80 x^{5}+160 x^{4}\\\hline\\&&&&&&{\color{Green}- 160 x^{4}}&+0 x^{3}&- 4 x^{2}&+0 x&+2&\frac{{\color{Green}- 160 x^{4}}}{{\color{Magenta}5 x}} = {\color{Green}- 32 x^{3}}\\&&&&&&-\phantom{- 160 x^{4}}&&&&&\\&&&&&&- 160 x^{4}&- 320 x^{3}&&&&{\color{Green}- 32 x^{3}} \left(5 x+10\right) = - 160 x^{4}- 320 x^{3}\\\hline\\&&&&&&&{\color{Blue}320 x^{3}}&- 4 x^{2}&+0 x&+2&\frac{{\color{Blue}320 x^{3}}}{{\color{Magenta}5 x}} = {\color{Blue}64 x^{2}}\\&&&&&&&-\phantom{320 x^{3}}&&&&\\&&&&&&&320 x^{3}&+640 x^{2}&&&{\color{Blue}64 x^{2}} \left(5 x+10\right) = 320 x^{3}+640 x^{2}\\\hline\\&&&&&&&&{\color{BlueViolet}- 644 x^{2}}&+0 x&+2&\frac{{\color{BlueViolet}- 644 x^{2}}}{{\color{Magenta}5 x}} = {\color{BlueViolet}- \frac{644 x}{5}}\\&&&&&&&&-\phantom{- 644 x^{2}}&&&\\&&&&&&&&- 644 x^{2}&- 1288 x&&{\color{BlueViolet}- \frac{644 x}{5}} \left(5 x+10\right) = - 644 x^{2}- 1288 x\\\hline\\&&&&&&&&&{\color{Purple}1288 x}&+2&\frac{{\color{Purple}1288 x}}{{\color{Magenta}5 x}} = {\color{Purple}\frac{1288}{5}}\\&&&&&&&&&-\phantom{1288 x}&&\\&&&&&&&&&1288 x&+2576&{\color{Purple}\frac{1288}{5}} \left(5 x+10\right) = 1288 x+2576\\\hline\\&&&&&&&&&&-2574&\end{array}$$

Therefore, $$$\frac{5 x^{9} - 4 x^{2} + 2}{5 x + 10} = \left(x^{8} - 2 x^{7} + 4 x^{6} - 8 x^{5} + 16 x^{4} - 32 x^{3} + 64 x^{2} - \frac{644 x}{5} + \frac{1288}{5}\right) + \frac{-2574}{5 x + 10}.$$$