Divide using synthetic division $$ ( x^{3}-5x^{2}+4x+6 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&1&-5&4&6\\& & -6& 66& \color{black}{-420} \\ \hline &\color{blue}{1}&\color{blue}{-11}&\color{blue}{70}&\color{orangered}{-414} \end{array} $$The solution is:
$$ \frac{ x^{3}-5x^{2}+4x+6 }{ x+6 } = \color{blue}{x^{2}-11x+70} \color{red}{~-~} \frac{ \color{red}{ 414 } }{ x+6 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-5&4&6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 1 }&-5&4&6\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-5&4&6\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrr}-6&1&\color{orangered}{ -5 }&4&6\\& & \color{orangered}{-6} & & \\ \hline &1&\color{orangered}{-11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 66 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-5&4&6\\& & -6& \color{blue}{66} & \\ \hline &1&\color{blue}{-11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 66 } = \color{orangered}{ 70 } $
$$ \begin{array}{c|rrrr}-6&1&-5&\color{orangered}{ 4 }&6\\& & -6& \color{orangered}{66} & \\ \hline &1&-11&\color{orangered}{70}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 70 } = \color{blue}{ -420 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-5&4&6\\& & -6& 66& \color{blue}{-420} \\ \hline &1&-11&\color{blue}{70}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -420 \right) } = \color{orangered}{ -414 } $
$$ \begin{array}{c|rrrr}-6&1&-5&4&\color{orangered}{ 6 }\\& & -6& 66& \color{orangered}{-420} \\ \hline &\color{blue}{1}&\color{blue}{-11}&\color{blue}{70}&\color{orangered}{-414} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-11x+70 } $ with a remainder of $ \color{red}{ -414 } $.