Divide using synthetic division $$ ( x^{4}-8x^{3}+18x^{2}+16x-40 ) \div ( -2x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&1&-8&18&16&-40\\& & -4& 48& -264& \color{black}{992} \\ \hline &\color{blue}{1}&\color{blue}{-12}&\color{blue}{66}&\color{blue}{-248}&\color{orangered}{952} \end{array} $$The solution is:
$$ \frac{ x^{4}-8x^{3}+18x^{2}+16x-40 }{ x+4 } = \color{blue}{x^{3}-12x^{2}+66x-248} ~+~ \frac{ \color{red}{ 952 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&-8&18&16&-40\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 1 }&-8&18&16&-40\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&-8&18&16&-40\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrr}-4&1&\color{orangered}{ -8 }&18&16&-40\\& & \color{orangered}{-4} & & & \\ \hline &1&\color{orangered}{-12}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&-8&18&16&-40\\& & -4& \color{blue}{48} & & \\ \hline &1&\color{blue}{-12}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ 48 } = \color{orangered}{ 66 } $
$$ \begin{array}{c|rrrrr}-4&1&-8&\color{orangered}{ 18 }&16&-40\\& & -4& \color{orangered}{48} & & \\ \hline &1&-12&\color{orangered}{66}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 66 } = \color{blue}{ -264 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&-8&18&16&-40\\& & -4& 48& \color{blue}{-264} & \\ \hline &1&-12&\color{blue}{66}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -264 \right) } = \color{orangered}{ -248 } $
$$ \begin{array}{c|rrrrr}-4&1&-8&18&\color{orangered}{ 16 }&-40\\& & -4& 48& \color{orangered}{-264} & \\ \hline &1&-12&66&\color{orangered}{-248}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -248 \right) } = \color{blue}{ 992 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&-8&18&16&-40\\& & -4& 48& -264& \color{blue}{992} \\ \hline &1&-12&66&\color{blue}{-248}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -40 } + \color{orangered}{ 992 } = \color{orangered}{ 952 } $
$$ \begin{array}{c|rrrrr}-4&1&-8&18&16&\color{orangered}{ -40 }\\& & -4& 48& -264& \color{orangered}{992} \\ \hline &\color{blue}{1}&\color{blue}{-12}&\color{blue}{66}&\color{blue}{-248}&\color{orangered}{952} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-12x^{2}+66x-248 } $ with a remainder of $ \color{red}{ 952 } $.