Divide $$$2 x^{3} + 15 x^{2} + 22 x - 11$$$ by $$$x^{2} + 8 x + 15$$$

Write the problem in the special format:

$$$\begin{array}{r|r}\hline\\x^{2}+8 x+15&2 x^{3}+15 x^{2}+22 x-11\end{array}$$$

Step 1

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

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

Multiply it by the divisor: $$$2 x \left(x^{2}+8 x+15\right) = 2 x^{3}+16 x^{2}+30 x$$$.

Subtract the dividend from the obtained result: $$$\left(2 x^{3}+15 x^{2}+22 x-11\right) - \left(2 x^{3}+16 x^{2}+30 x\right) = - x^{2}- 8 x-11$$$.

$$\begin{array}{r|rrrr:c}&{\color{Chartreuse}2 x}&&&&\\\hline\\{\color{Magenta}x^{2}}+8 x+15&{\color{Chartreuse}2 x^{3}}&+15 x^{2}&+22 x&-11&\frac{{\color{Chartreuse}2 x^{3}}}{{\color{Magenta}x^{2}}} = {\color{Chartreuse}2 x}\\&-\phantom{2 x^{3}}&&&&\\&2 x^{3}&+16 x^{2}&+30 x&&{\color{Chartreuse}2 x} \left(x^{2}+8 x+15\right) = 2 x^{3}+16 x^{2}+30 x\\\hline\\&&- x^{2}&- 8 x&-11&\end{array}$$

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{- x^{2}}{x^{2}} = -1$$$.

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

Multiply it by the divisor: $$$- \left(x^{2}+8 x+15\right) = - x^{2}- 8 x-15$$$.

Subtract the remainder from the obtained result: $$$\left(- x^{2}- 8 x-11\right) - \left(- x^{2}- 8 x-15\right) = 4$$$.

$$\begin{array}{r|rrrr:c}&2 x&{\color{GoldenRod}-1}&&&\\\hline\\{\color{Magenta}x^{2}}+8 x+15&2 x^{3}&+15 x^{2}&+22 x&-11&\\&-\phantom{2 x^{3}}&&&&\\&2 x^{3}&+16 x^{2}&+30 x&&\\\hline\\&&{\color{GoldenRod}- x^{2}}&- 8 x&-11&\frac{{\color{GoldenRod}- x^{2}}}{{\color{Magenta}x^{2}}} = {\color{GoldenRod}-1}\\&&-\phantom{- x^{2}}&&&\\&&- x^{2}&- 8 x&-15&{\color{GoldenRod}-1} \left(x^{2}+8 x+15\right) = - x^{2}- 8 x-15\\\hline\\&&&&4&\end{array}$$

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

The resulting table is shown once more:

$$\begin{array}{r|rrrr:c}&{\color{Chartreuse}2 x}&{\color{GoldenRod}-1}&&&\text{Hints}\\\hline\\{\color{Magenta}x^{2}}+8 x+15&{\color{Chartreuse}2 x^{3}}&+15 x^{2}&+22 x&-11&\frac{{\color{Chartreuse}2 x^{3}}}{{\color{Magenta}x^{2}}} = {\color{Chartreuse}2 x}\\&-\phantom{2 x^{3}}&&&&\\&2 x^{3}&+16 x^{2}&+30 x&&{\color{Chartreuse}2 x} \left(x^{2}+8 x+15\right) = 2 x^{3}+16 x^{2}+30 x\\\hline\\&&{\color{GoldenRod}- x^{2}}&- 8 x&-11&\frac{{\color{GoldenRod}- x^{2}}}{{\color{Magenta}x^{2}}} = {\color{GoldenRod}-1}\\&&-\phantom{- x^{2}}&&&\\&&- x^{2}&- 8 x&-15&{\color{GoldenRod}-1} \left(x^{2}+8 x+15\right) = - x^{2}- 8 x-15\\\hline\\&&&&4&\end{array}$$

Therefore, $$$\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15} = \left(2 x - 1\right) + \frac{4}{x^{2} + 8 x + 15}$$$.