Divide using synthetic division $$ ( -x^{5}+x^{4}+16x-16 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}2&-1&1&0&0&16&-16\\& & -2& -2& -4& -8& \color{black}{16} \\ \hline &\color{blue}{-1}&\color{blue}{-1}&\color{blue}{-2}&\color{blue}{-4}&\color{blue}{8}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -x^{5}+x^{4}+16x-16 }{ x-2 } = \color{blue}{-x^{4}-x^{3}-2x^{2}-4x+8} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-1&1&0&0&16&-16\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}2&\color{orangered}{ -1 }&1&0&0&16&-16\\& & & & & & \\ \hline &\color{orangered}{-1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-1&1&0&0&16&-16\\& & \color{blue}{-2} & & & & \\ \hline &\color{blue}{-1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrr}2&-1&\color{orangered}{ 1 }&0&0&16&-16\\& & \color{orangered}{-2} & & & & \\ \hline &-1&\color{orangered}{-1}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-1&1&0&0&16&-16\\& & -2& \color{blue}{-2} & & & \\ \hline &-1&\color{blue}{-1}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrr}2&-1&1&\color{orangered}{ 0 }&0&16&-16\\& & -2& \color{orangered}{-2} & & & \\ \hline &-1&-1&\color{orangered}{-2}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-1&1&0&0&16&-16\\& & -2& -2& \color{blue}{-4} & & \\ \hline &-1&-1&\color{blue}{-2}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}2&-1&1&0&\color{orangered}{ 0 }&16&-16\\& & -2& -2& \color{orangered}{-4} & & \\ \hline &-1&-1&-2&\color{orangered}{-4}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-1&1&0&0&16&-16\\& & -2& -2& -4& \color{blue}{-8} & \\ \hline &-1&-1&-2&\color{blue}{-4}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrr}2&-1&1&0&0&\color{orangered}{ 16 }&-16\\& & -2& -2& -4& \color{orangered}{-8} & \\ \hline &-1&-1&-2&-4&\color{orangered}{8}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 8 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&-1&1&0&0&16&-16\\& & -2& -2& -4& -8& \color{blue}{16} \\ \hline &-1&-1&-2&-4&\color{blue}{8}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 16 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}2&-1&1&0&0&16&\color{orangered}{ -16 }\\& & -2& -2& -4& -8& \color{orangered}{16} \\ \hline &\color{blue}{-1}&\color{blue}{-1}&\color{blue}{-2}&\color{blue}{-4}&\color{blue}{8}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{4}-x^{3}-2x^{2}-4x+8 } $ with a remainder of $ \color{red}{ 0 } $.