Divide $$$x^{5} - 4 x^{4} - 10 x^{3} + 48 x^{2} - 17 x - 26$$$ by $$$x^{2} - 3 x - 2$$$

Write the problem in the special format:

$$$\begin{array}{r|r}\hline\\x^{2}- 3 x-2&x^{5}- 4 x^{4}- 10 x^{3}+48 x^{2}- 17 x-26\end{array}$$$

Step 1

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

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

Multiply it by the divisor: $$$x^{3} \left(x^{2}- 3 x-2\right) = x^{5}- 3 x^{4}- 2 x^{3}$$$.

Subtract the dividend from the obtained result: $$$\left(x^{5}- 4 x^{4}- 10 x^{3}+48 x^{2}- 17 x-26\right) - \left(x^{5}- 3 x^{4}- 2 x^{3}\right) = - x^{4}- 8 x^{3}+48 x^{2}- 17 x-26.$$$

$$\begin{array}{r|rrrrrr:c}&{\color{DeepPink}x^{3}}&&&&&&\\\hline\\{\color{Magenta}x^{2}}- 3 x-2&{\color{DeepPink}x^{5}}&- 4 x^{4}&- 10 x^{3}&+48 x^{2}&- 17 x&-26&\frac{{\color{DeepPink}x^{5}}}{{\color{Magenta}x^{2}}} = {\color{DeepPink}x^{3}}\\&-\phantom{x^{5}}&&&&&&\\&x^{5}&- 3 x^{4}&- 2 x^{3}&&&&{\color{DeepPink}x^{3}} \left(x^{2}- 3 x-2\right) = x^{5}- 3 x^{4}- 2 x^{3}\\\hline\\&&- x^{4}&- 8 x^{3}&+48 x^{2}&- 17 x&-26&\end{array}$$

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- x^{4}}{x^{2}} = - x^{2}$$$.

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

Multiply it by the divisor: $$$- x^{2} \left(x^{2}- 3 x-2\right) = - x^{4}+3 x^{3}+2 x^{2}$$$.

Subtract the remainder from the obtained result: $$$\left(- x^{4}- 8 x^{3}+48 x^{2}- 17 x-26\right) - \left(- x^{4}+3 x^{3}+2 x^{2}\right) = - 11 x^{3}+46 x^{2}- 17 x-26.$$$

$$\begin{array}{r|rrrrrr:c}&x^{3}&{\color{Brown}- x^{2}}&&&&&\\\hline\\{\color{Magenta}x^{2}}- 3 x-2&x^{5}&- 4 x^{4}&- 10 x^{3}&+48 x^{2}&- 17 x&-26&\\&-\phantom{x^{5}}&&&&&&\\&x^{5}&- 3 x^{4}&- 2 x^{3}&&&&\\\hline\\&&{\color{Brown}- x^{4}}&- 8 x^{3}&+48 x^{2}&- 17 x&-26&\frac{{\color{Brown}- x^{4}}}{{\color{Magenta}x^{2}}} = {\color{Brown}- x^{2}}\\&&-\phantom{- x^{4}}&&&&&\\&&- x^{4}&+3 x^{3}&+2 x^{2}&&&{\color{Brown}- x^{2}} \left(x^{2}- 3 x-2\right) = - x^{4}+3 x^{3}+2 x^{2}\\\hline\\&&&- 11 x^{3}&+46 x^{2}&- 17 x&-26&\end{array}$$

Step 3

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- 11 x^{3}}{x^{2}} = - 11 x$$$.

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

Multiply it by the divisor: $$$- 11 x \left(x^{2}- 3 x-2\right) = - 11 x^{3}+33 x^{2}+22 x$$$.

Subtract the remainder from the obtained result: $$$\left(- 11 x^{3}+46 x^{2}- 17 x-26\right) - \left(- 11 x^{3}+33 x^{2}+22 x\right) = 13 x^{2}- 39 x-26$$$.

$$\begin{array}{r|rrrrrr:c}&x^{3}&- x^{2}&{\color{Fuchsia}- 11 x}&&&&\\\hline\\{\color{Magenta}x^{2}}- 3 x-2&x^{5}&- 4 x^{4}&- 10 x^{3}&+48 x^{2}&- 17 x&-26&\\&-\phantom{x^{5}}&&&&&&\\&x^{5}&- 3 x^{4}&- 2 x^{3}&&&&\\\hline\\&&- x^{4}&- 8 x^{3}&+48 x^{2}&- 17 x&-26&\\&&-\phantom{- x^{4}}&&&&&\\&&- x^{4}&+3 x^{3}&+2 x^{2}&&&\\\hline\\&&&{\color{Fuchsia}- 11 x^{3}}&+46 x^{2}&- 17 x&-26&\frac{{\color{Fuchsia}- 11 x^{3}}}{{\color{Magenta}x^{2}}} = {\color{Fuchsia}- 11 x}\\&&&-\phantom{- 11 x^{3}}&&&&\\&&&- 11 x^{3}&+33 x^{2}&+22 x&&{\color{Fuchsia}- 11 x} \left(x^{2}- 3 x-2\right) = - 11 x^{3}+33 x^{2}+22 x\\\hline\\&&&&13 x^{2}&- 39 x&-26&\end{array}$$

Step 4

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{13 x^{2}}{x^{2}} = 13$$$.

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

Multiply it by the divisor: $$$13 \left(x^{2}- 3 x-2\right) = 13 x^{2}- 39 x-26$$$.

Subtract the remainder from the obtained result: $$$\left(13 x^{2}- 39 x-26\right) - \left(13 x^{2}- 39 x-26\right) = $$$.

$$\begin{array}{r|rrrrrr:c}&x^{3}&- x^{2}&- 11 x&{\color{DarkCyan}+13}&&&\\\hline\\{\color{Magenta}x^{2}}- 3 x-2&x^{5}&- 4 x^{4}&- 10 x^{3}&+48 x^{2}&- 17 x&-26&\\&-\phantom{x^{5}}&&&&&&\\&x^{5}&- 3 x^{4}&- 2 x^{3}&&&&\\\hline\\&&- x^{4}&- 8 x^{3}&+48 x^{2}&- 17 x&-26&\\&&-\phantom{- x^{4}}&&&&&\\&&- x^{4}&+3 x^{3}&+2 x^{2}&&&\\\hline\\&&&- 11 x^{3}&+46 x^{2}&- 17 x&-26&\\&&&-\phantom{- 11 x^{3}}&&&&\\&&&- 11 x^{3}&+33 x^{2}&+22 x&&\\\hline\\&&&&{\color{DarkCyan}13 x^{2}}&- 39 x&-26&\frac{{\color{DarkCyan}13 x^{2}}}{{\color{Magenta}x^{2}}} = {\color{DarkCyan}13}\\&&&&-\phantom{13 x^{2}}&&&\\&&&&13 x^{2}&- 39 x&-26&{\color{DarkCyan}13} \left(x^{2}- 3 x-2\right) = 13 x^{2}- 39 x-26\\\hline\\&&&&&&0&\end{array}$$

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

The resulting table is shown once more:

$$\begin{array}{r|rrrrrr:c}&{\color{DeepPink}x^{3}}&{\color{Brown}- x^{2}}&{\color{Fuchsia}- 11 x}&{\color{DarkCyan}+13}&&&\text{Hints}\\\hline\\{\color{Magenta}x^{2}}- 3 x-2&{\color{DeepPink}x^{5}}&- 4 x^{4}&- 10 x^{3}&+48 x^{2}&- 17 x&-26&\frac{{\color{DeepPink}x^{5}}}{{\color{Magenta}x^{2}}} = {\color{DeepPink}x^{3}}\\&-\phantom{x^{5}}&&&&&&\\&x^{5}&- 3 x^{4}&- 2 x^{3}&&&&{\color{DeepPink}x^{3}} \left(x^{2}- 3 x-2\right) = x^{5}- 3 x^{4}- 2 x^{3}\\\hline\\&&{\color{Brown}- x^{4}}&- 8 x^{3}&+48 x^{2}&- 17 x&-26&\frac{{\color{Brown}- x^{4}}}{{\color{Magenta}x^{2}}} = {\color{Brown}- x^{2}}\\&&-\phantom{- x^{4}}&&&&&\\&&- x^{4}&+3 x^{3}&+2 x^{2}&&&{\color{Brown}- x^{2}} \left(x^{2}- 3 x-2\right) = - x^{4}+3 x^{3}+2 x^{2}\\\hline\\&&&{\color{Fuchsia}- 11 x^{3}}&+46 x^{2}&- 17 x&-26&\frac{{\color{Fuchsia}- 11 x^{3}}}{{\color{Magenta}x^{2}}} = {\color{Fuchsia}- 11 x}\\&&&-\phantom{- 11 x^{3}}&&&&\\&&&- 11 x^{3}&+33 x^{2}&+22 x&&{\color{Fuchsia}- 11 x} \left(x^{2}- 3 x-2\right) = - 11 x^{3}+33 x^{2}+22 x\\\hline\\&&&&{\color{DarkCyan}13 x^{2}}&- 39 x&-26&\frac{{\color{DarkCyan}13 x^{2}}}{{\color{Magenta}x^{2}}} = {\color{DarkCyan}13}\\&&&&-\phantom{13 x^{2}}&&&\\&&&&13 x^{2}&- 39 x&-26&{\color{DarkCyan}13} \left(x^{2}- 3 x-2\right) = 13 x^{2}- 39 x-26\\\hline\\&&&&&&0&\end{array}$$

Therefore, $$$\frac{x^{5} - 4 x^{4} - 10 x^{3} + 48 x^{2} - 17 x - 26}{x^{2} - 3 x - 2} = \left(x^{3} - x^{2} - 11 x + 13\right) + \frac{0}{x^{2} - 3 x - 2} = x^{3} - x^{2} - 11 x + 13.$$$