Divide using synthetic division $$ ( -x^{3}-8x-35 ) \div ( x-5 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&-1&0&-8&-35\\& & -5& -25& \color{black}{-165} \\ \hline &\color{blue}{-1}&\color{blue}{-5}&\color{blue}{-33}&\color{orangered}{-200} \end{array} $$The solution is:
$$ \dfrac{ -x^{3}-8x-35 }{ x-5 } = \color{blue}{-x^{2}-5x-33} \color{red}{~-~} \dfrac{ \color{red}{ 200 } }{ x-5 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&0&-8&-35\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ -1 }&0&-8&-35\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&0&-8&-35\\& & \color{blue}{-5} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}5&-1&\color{orangered}{ 0 }&-8&-35\\& & \color{orangered}{-5} & & \\ \hline &-1&\color{orangered}{-5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&0&-8&-35\\& & -5& \color{blue}{-25} & \\ \hline &-1&\color{blue}{-5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ -33 } $
$$ \begin{array}{c|rrrr}5&-1&0&\color{orangered}{ -8 }&-35\\& & -5& \color{orangered}{-25} & \\ \hline &-1&-5&\color{orangered}{-33}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -33 \right) } = \color{blue}{ -165 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&0&-8&-35\\& & -5& -25& \color{blue}{-165} \\ \hline &-1&-5&\color{blue}{-33}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -35 } + \color{orangered}{ \left( -165 \right) } = \color{orangered}{ -200 } $
$$ \begin{array}{c|rrrr}5&-1&0&-8&\color{orangered}{ -35 }\\& & -5& -25& \color{orangered}{-165} \\ \hline &\color{blue}{-1}&\color{blue}{-5}&\color{blue}{-33}&\color{orangered}{-200} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}-5x-33 } $ with a remainder of $ \color{red}{ -200 } $.