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