Divide using synthetic division $$ ( -x^{3}+6x^{2}-12x-16 ) \div ( x+8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-8&-1&6&-12&-16\\& & 8& -112& \color{black}{992} \\ \hline &\color{blue}{-1}&\color{blue}{14}&\color{blue}{-124}&\color{orangered}{976} \end{array} $$The solution is:
$$ \frac{ -x^{3}+6x^{2}-12x-16 }{ x+8 } = \color{blue}{-x^{2}+14x-124} ~+~ \frac{ \color{red}{ 976 } }{ x+8 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 8 = 0 $ ( $ x = \color{blue}{ -8 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&-1&6&-12&-16\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-8&\color{orangered}{ -1 }&6&-12&-16\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&-1&6&-12&-16\\& & \color{blue}{8} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 8 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrr}-8&-1&\color{orangered}{ 6 }&-12&-16\\& & \color{orangered}{8} & & \\ \hline &-1&\color{orangered}{14}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 14 } = \color{blue}{ -112 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&-1&6&-12&-16\\& & 8& \color{blue}{-112} & \\ \hline &-1&\color{blue}{14}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ \left( -112 \right) } = \color{orangered}{ -124 } $
$$ \begin{array}{c|rrrr}-8&-1&6&\color{orangered}{ -12 }&-16\\& & 8& \color{orangered}{-112} & \\ \hline &-1&14&\color{orangered}{-124}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -124 \right) } = \color{blue}{ 992 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-8}&-1&6&-12&-16\\& & 8& -112& \color{blue}{992} \\ \hline &-1&14&\color{blue}{-124}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 992 } = \color{orangered}{ 976 } $
$$ \begin{array}{c|rrrr}-8&-1&6&-12&\color{orangered}{ -16 }\\& & 8& -112& \color{orangered}{992} \\ \hline &\color{blue}{-1}&\color{blue}{14}&\color{blue}{-124}&\color{orangered}{976} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}+14x-124 } $ with a remainder of $ \color{red}{ 976 } $.