Divide using synthetic division $$ ( x^{4}-3x^{3}-3x^{2}-7x+6 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-6&1&-3&-3&-7&6\\& & -6& 54& -306& \color{black}{1878} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{51}&\color{blue}{-313}&\color{orangered}{1884} \end{array} $$The solution is:
$$ \frac{ x^{4}-3x^{3}-3x^{2}-7x+6 }{ x+6 } = \color{blue}{x^{3}-9x^{2}+51x-313} ~+~ \frac{ \color{red}{ 1884 } }{ x+6 } $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&-3&-7&6\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-6&\color{orangered}{ 1 }&-3&-3&-7&6\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 1 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&-3&-7&6\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-6&1&\color{orangered}{ -3 }&-3&-7&6\\& & \color{orangered}{-6} & & & \\ \hline &1&\color{orangered}{-9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&-3&-7&6\\& & -6& \color{blue}{54} & & \\ \hline &1&\color{blue}{-9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 54 } = \color{orangered}{ 51 } $
$$ \begin{array}{c|rrrrr}-6&1&-3&\color{orangered}{ -3 }&-7&6\\& & -6& \color{orangered}{54} & & \\ \hline &1&-9&\color{orangered}{51}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 51 } = \color{blue}{ -306 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&-3&-7&6\\& & -6& 54& \color{blue}{-306} & \\ \hline &1&-9&\color{blue}{51}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -306 \right) } = \color{orangered}{ -313 } $
$$ \begin{array}{c|rrrrr}-6&1&-3&-3&\color{orangered}{ -7 }&6\\& & -6& 54& \color{orangered}{-306} & \\ \hline &1&-9&51&\color{orangered}{-313}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -313 \right) } = \color{blue}{ 1878 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&-3&-7&6\\& & -6& 54& -306& \color{blue}{1878} \\ \hline &1&-9&51&\color{blue}{-313}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 1878 } = \color{orangered}{ 1884 } $
$$ \begin{array}{c|rrrrr}-6&1&-3&-3&-7&\color{orangered}{ 6 }\\& & -6& 54& -306& \color{orangered}{1878} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{51}&\color{blue}{-313}&\color{orangered}{1884} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-9x^{2}+51x-313 } $ with a remainder of $ \color{red}{ 1884 } $.