Divide using synthetic division $$ ( -x^{5}+x^{4}+16x-16 ) \div ( x+1 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-1&-1&1&0&0&16&-16\\& & 1& -2& 2& -2& \color{black}{-14} \\ \hline &\color{blue}{-1}&\color{blue}{2}&\color{blue}{-2}&\color{blue}{2}&\color{blue}{14}&\color{orangered}{-30} \end{array} $$The solution is:
$$ \dfrac{ -x^{5}+x^{4}+16x-16 }{ x+1 } = \color{blue}{-x^{4}+2x^{3}-2x^{2}+2x+14} \color{red}{~-~} \dfrac{ \color{red}{ 30 } }{ x+1 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&1&0&0&16&-16\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-1&\color{orangered}{ -1 }&1&0&0&16&-16\\& & & & & & \\ \hline &\color{orangered}{-1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&1&0&0&16&-16\\& & \color{blue}{1} & & & & \\ \hline &\color{blue}{-1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 1 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrr}-1&-1&\color{orangered}{ 1 }&0&0&16&-16\\& & \color{orangered}{1} & & & & \\ \hline &-1&\color{orangered}{2}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 2 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&1&0&0&16&-16\\& & 1& \color{blue}{-2} & & & \\ \hline &-1&\color{blue}{2}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrr}-1&-1&1&\color{orangered}{ 0 }&0&16&-16\\& & 1& \color{orangered}{-2} & & & \\ \hline &-1&2&\color{orangered}{-2}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&1&0&0&16&-16\\& & 1& -2& \color{blue}{2} & & \\ \hline &-1&2&\color{blue}{-2}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrr}-1&-1&1&0&\color{orangered}{ 0 }&16&-16\\& & 1& -2& \color{orangered}{2} & & \\ \hline &-1&2&-2&\color{orangered}{2}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 2 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&1&0&0&16&-16\\& & 1& -2& 2& \color{blue}{-2} & \\ \hline &-1&2&-2&\color{blue}{2}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrrrr}-1&-1&1&0&0&\color{orangered}{ 16 }&-16\\& & 1& -2& 2& \color{orangered}{-2} & \\ \hline &-1&2&-2&2&\color{orangered}{14}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 14 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-1&1&0&0&16&-16\\& & 1& -2& 2& -2& \color{blue}{-14} \\ \hline &-1&2&-2&2&\color{blue}{14}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -30 } $
$$ \begin{array}{c|rrrrrr}-1&-1&1&0&0&16&\color{orangered}{ -16 }\\& & 1& -2& 2& -2& \color{orangered}{-14} \\ \hline &\color{blue}{-1}&\color{blue}{2}&\color{blue}{-2}&\color{blue}{2}&\color{blue}{14}&\color{orangered}{-30} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{4}+2x^{3}-2x^{2}+2x+14 } $ with a remainder of $ \color{red}{ -30 } $.