Divide $$$6 x^{3} + 11 x^{2} - 8 x - 20$$$ by $$$6 x^{2} - x - 12$$$

Write the problem in the special format:

$$$\begin{array}{r|r}\hline\\6 x^{2}- x-12&6 x^{3}+11 x^{2}- 8 x-20\end{array}$$$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$$\frac{6 x^{3}}{6 x^{2}} = x$$$.

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

Multiply it by the divisor: $$$x \left(6 x^{2}- x-12\right) = 6 x^{3}- x^{2}- 12 x$$$.

Subtract the dividend from the obtained result: $$$\left(6 x^{3}+11 x^{2}- 8 x-20\right) - \left(6 x^{3}- x^{2}- 12 x\right) = 12 x^{2}+4 x-20$$$.

$$\begin{array}{r|rrrr:c}&{\color{BlueViolet}x}&&&&\\\hline\\{\color{Magenta}6 x^{2}}- x-12&{\color{BlueViolet}6 x^{3}}&+11 x^{2}&- 8 x&-20&\frac{{\color{BlueViolet}6 x^{3}}}{{\color{Magenta}6 x^{2}}} = {\color{BlueViolet}x}\\&-\phantom{6 x^{3}}&&&&\\&6 x^{3}&- x^{2}&- 12 x&&{\color{BlueViolet}x} \left(6 x^{2}- x-12\right) = 6 x^{3}- x^{2}- 12 x\\\hline\\&&12 x^{2}&+4 x&-20&\end{array}$$

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $$$\frac{12 x^{2}}{6 x^{2}} = 2$$$.

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

Multiply it by the divisor: $$$2 \left(6 x^{2}- x-12\right) = 12 x^{2}- 2 x-24$$$.

Subtract the remainder from the obtained result: $$$\left(12 x^{2}+4 x-20\right) - \left(12 x^{2}- 2 x-24\right) = 6 x+4$$$.

$$\begin{array}{r|rrrr:c}&x&{\color{Fuchsia}+2}&&&\\\hline\\{\color{Magenta}6 x^{2}}- x-12&6 x^{3}&+11 x^{2}&- 8 x&-20&\\&-\phantom{6 x^{3}}&&&&\\&6 x^{3}&- x^{2}&- 12 x&&\\\hline\\&&{\color{Fuchsia}12 x^{2}}&+4 x&-20&\frac{{\color{Fuchsia}12 x^{2}}}{{\color{Magenta}6 x^{2}}} = {\color{Fuchsia}2}\\&&-\phantom{12 x^{2}}&&&\\&&12 x^{2}&- 2 x&-24&{\color{Fuchsia}2} \left(6 x^{2}- x-12\right) = 12 x^{2}- 2 x-24\\\hline\\&&&6 x&+4&\end{array}$$

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

The resulting table is shown once more:

$$\begin{array}{r|rrrr:c}&{\color{BlueViolet}x}&{\color{Fuchsia}+2}&&&\text{Hints}\\\hline\\{\color{Magenta}6 x^{2}}- x-12&{\color{BlueViolet}6 x^{3}}&+11 x^{2}&- 8 x&-20&\frac{{\color{BlueViolet}6 x^{3}}}{{\color{Magenta}6 x^{2}}} = {\color{BlueViolet}x}\\&-\phantom{6 x^{3}}&&&&\\&6 x^{3}&- x^{2}&- 12 x&&{\color{BlueViolet}x} \left(6 x^{2}- x-12\right) = 6 x^{3}- x^{2}- 12 x\\\hline\\&&{\color{Fuchsia}12 x^{2}}&+4 x&-20&\frac{{\color{Fuchsia}12 x^{2}}}{{\color{Magenta}6 x^{2}}} = {\color{Fuchsia}2}\\&&-\phantom{12 x^{2}}&&&\\&&12 x^{2}&- 2 x&-24&{\color{Fuchsia}2} \left(6 x^{2}- x-12\right) = 12 x^{2}- 2 x-24\\\hline\\&&&6 x&+4&\end{array}$$

Therefore, $$$\frac{6 x^{3} + 11 x^{2} - 8 x - 20}{6 x^{2} - x - 12} = \left(x + 2\right) + \frac{6 x + 4}{6 x^{2} - x - 12}$$$.