Divide using synthetic division $$ ( -x^{3}-6x^{2}+9x+8 ) \div ( x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-7&-1&-6&9&8\\& & 7& -7& \color{black}{-14} \\ \hline &\color{blue}{-1}&\color{blue}{1}&\color{blue}{2}&\color{orangered}{-6} \end{array} $$The solution is:
$$ \frac{ -x^{3}-6x^{2}+9x+8 }{ x+7 } = \color{blue}{-x^{2}+x+2} \color{red}{~-~} \frac{ \color{red}{ 6 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&-1&-6&9&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-7&\color{orangered}{ -1 }&-6&9&8\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 7 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&-1&-6&9&8\\& & \color{blue}{7} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 7 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}-7&-1&\color{orangered}{ -6 }&9&8\\& & \color{orangered}{7} & & \\ \hline &-1&\color{orangered}{1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 1 } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&-1&-6&9&8\\& & 7& \color{blue}{-7} & \\ \hline &-1&\color{blue}{1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}-7&-1&-6&\color{orangered}{ 9 }&8\\& & 7& \color{orangered}{-7} & \\ \hline &-1&1&\color{orangered}{2}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 2 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&-1&-6&9&8\\& & 7& -7& \color{blue}{-14} \\ \hline &-1&1&\color{blue}{2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}-7&-1&-6&9&\color{orangered}{ 8 }\\& & 7& -7& \color{orangered}{-14} \\ \hline &\color{blue}{-1}&\color{blue}{1}&\color{blue}{2}&\color{orangered}{-6} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}+x+2 } $ with a remainder of $ \color{red}{ -6 } $.