Divide using synthetic division $$ ( 3x^{3}+18x^{2}-6x-36 ) \div ( x+6 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&3&18&-6&-36\\& & -18& 0& \color{black}{36} \\ \hline &\color{blue}{3}&\color{blue}{0}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ 3x^{3}+18x^{2}-6x-36 }{ x+6 } = \color{blue}{3x^{2}-6} $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&3&18&-6&-36\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 3 }&18&-6&-36\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 3 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&3&18&-6&-36\\& & \color{blue}{-18} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-6&3&\color{orangered}{ 18 }&-6&-36\\& & \color{orangered}{-18} & & \\ \hline &3&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&3&18&-6&-36\\& & -18& \color{blue}{0} & \\ \hline &3&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 0 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}-6&3&18&\color{orangered}{ -6 }&-36\\& & -18& \color{orangered}{0} & \\ \hline &3&0&\color{orangered}{-6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&3&18&-6&-36\\& & -18& 0& \color{blue}{36} \\ \hline &3&0&\color{blue}{-6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ 36 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-6&3&18&-6&\color{orangered}{ -36 }\\& & -18& 0& \color{orangered}{36} \\ \hline &\color{blue}{3}&\color{blue}{0}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-6 } $ with a remainder of $ \color{red}{ 0 } $.