Divide $$$2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12$$$ by $$$x^{2} - 4 x - 12$$$

Write the problem in the special format:

$$$\begin{array}{r|r}\hline\\x^{2}- 4 x-12&2 x^{4}- 3 x^{3}- 15 x^{2}+32 x-12\end{array}$$$

Step 1

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

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

Multiply it by the divisor: $$$2 x^{2} \left(x^{2}- 4 x-12\right) = 2 x^{4}- 8 x^{3}- 24 x^{2}$$$.

Subtract the dividend from the obtained result: $$$\left(2 x^{4}- 3 x^{3}- 15 x^{2}+32 x-12\right) - \left(2 x^{4}- 8 x^{3}- 24 x^{2}\right) = 5 x^{3}+9 x^{2}+32 x-12.$$$

$$\begin{array}{r|rrrrr:c}&{\color{Peru}2 x^{2}}&&&&&\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&{\color{Peru}2 x^{4}}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\frac{{\color{Peru}2 x^{4}}}{{\color{Magenta}x^{2}}} = {\color{Peru}2 x^{2}}\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&{\color{Peru}2 x^{2}} \left(x^{2}- 4 x-12\right) = 2 x^{4}- 8 x^{3}- 24 x^{2}\\\hline\\&&5 x^{3}&+9 x^{2}&+32 x&-12&\end{array}$$

Step 2

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

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

Multiply it by the divisor: $$$5 x \left(x^{2}- 4 x-12\right) = 5 x^{3}- 20 x^{2}- 60 x$$$.

Subtract the remainder from the obtained result: $$$\left(5 x^{3}+9 x^{2}+32 x-12\right) - \left(5 x^{3}- 20 x^{2}- 60 x\right) = 29 x^{2}+92 x-12$$$.

$$\begin{array}{r|rrrrr:c}&2 x^{2}&{\color{DeepPink}+5 x}&&&&\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&2 x^{4}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&\\\hline\\&&{\color{DeepPink}5 x^{3}}&+9 x^{2}&+32 x&-12&\frac{{\color{DeepPink}5 x^{3}}}{{\color{Magenta}x^{2}}} = {\color{DeepPink}5 x}\\&&-\phantom{5 x^{3}}&&&&\\&&5 x^{3}&- 20 x^{2}&- 60 x&&{\color{DeepPink}5 x} \left(x^{2}- 4 x-12\right) = 5 x^{3}- 20 x^{2}- 60 x\\\hline\\&&&29 x^{2}&+92 x&-12&\end{array}$$

Step 3

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{29 x^{2}}{x^{2}} = 29$$$.

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

Multiply it by the divisor: $$$29 \left(x^{2}- 4 x-12\right) = 29 x^{2}- 116 x-348$$$.

Subtract the remainder from the obtained result: $$$\left(29 x^{2}+92 x-12\right) - \left(29 x^{2}- 116 x-348\right) = 208 x+336$$$.

$$\begin{array}{r|rrrrr:c}&2 x^{2}&+5 x&{\color{DarkMagenta}+29}&&&\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&2 x^{4}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&\\\hline\\&&5 x^{3}&+9 x^{2}&+32 x&-12&\\&&-\phantom{5 x^{3}}&&&&\\&&5 x^{3}&- 20 x^{2}&- 60 x&&\\\hline\\&&&{\color{DarkMagenta}29 x^{2}}&+92 x&-12&\frac{{\color{DarkMagenta}29 x^{2}}}{{\color{Magenta}x^{2}}} = {\color{DarkMagenta}29}\\&&&-\phantom{29 x^{2}}&&&\\&&&29 x^{2}&- 116 x&-348&{\color{DarkMagenta}29} \left(x^{2}- 4 x-12\right) = 29 x^{2}- 116 x-348\\\hline\\&&&&208 x&+336&\end{array}$$

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

The resulting table is shown once more:

$$\begin{array}{r|rrrrr:c}&{\color{Peru}2 x^{2}}&{\color{DeepPink}+5 x}&{\color{DarkMagenta}+29}&&&\text{Hints}\\\hline\\{\color{Magenta}x^{2}}- 4 x-12&{\color{Peru}2 x^{4}}&- 3 x^{3}&- 15 x^{2}&+32 x&-12&\frac{{\color{Peru}2 x^{4}}}{{\color{Magenta}x^{2}}} = {\color{Peru}2 x^{2}}\\&-\phantom{2 x^{4}}&&&&&\\&2 x^{4}&- 8 x^{3}&- 24 x^{2}&&&{\color{Peru}2 x^{2}} \left(x^{2}- 4 x-12\right) = 2 x^{4}- 8 x^{3}- 24 x^{2}\\\hline\\&&{\color{DeepPink}5 x^{3}}&+9 x^{2}&+32 x&-12&\frac{{\color{DeepPink}5 x^{3}}}{{\color{Magenta}x^{2}}} = {\color{DeepPink}5 x}\\&&-\phantom{5 x^{3}}&&&&\\&&5 x^{3}&- 20 x^{2}&- 60 x&&{\color{DeepPink}5 x} \left(x^{2}- 4 x-12\right) = 5 x^{3}- 20 x^{2}- 60 x\\\hline\\&&&{\color{DarkMagenta}29 x^{2}}&+92 x&-12&\frac{{\color{DarkMagenta}29 x^{2}}}{{\color{Magenta}x^{2}}} = {\color{DarkMagenta}29}\\&&&-\phantom{29 x^{2}}&&&\\&&&29 x^{2}&- 116 x&-348&{\color{DarkMagenta}29} \left(x^{2}- 4 x-12\right) = 29 x^{2}- 116 x-348\\\hline\\&&&&208 x&+336&\end{array}$$

Therefore, $$$\frac{2 x^{4} - 3 x^{3} - 15 x^{2} + 32 x - 12}{x^{2} - 4 x - 12} = \left(2 x^{2} + 5 x + 29\right) + \frac{208 x + 336}{x^{2} - 4 x - 12}.$$$