Divide using synthetic division $$ ( -11x^{4}+2x^{3}-8x^{2}-4 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-1&-11&2&-8&0&-4\\& & 11& -13& 21& \color{black}{-21} \\ \hline &\color{blue}{-11}&\color{blue}{13}&\color{blue}{-21}&\color{blue}{21}&\color{orangered}{-25} \end{array} $$The solution is:
$$ \frac{ -11x^{4}+2x^{3}-8x^{2}-4 }{ x+1 } = \color{blue}{-11x^{3}+13x^{2}-21x+21} \color{red}{~-~} \frac{ \color{red}{ 25 } }{ x+1 } $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&-11&2&-8&0&-4\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ -11 }&2&-8&0&-4\\& & & & & \\ \hline &\color{orangered}{-11}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 11 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&-11&2&-8&0&-4\\& & \color{blue}{11} & & & \\ \hline &\color{blue}{-11}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 11 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrrr}-1&-11&\color{orangered}{ 2 }&-8&0&-4\\& & \color{orangered}{11} & & & \\ \hline &-11&\color{orangered}{13}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 13 } = \color{blue}{ -13 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&-11&2&-8&0&-4\\& & 11& \color{blue}{-13} & & \\ \hline &-11&\color{blue}{13}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -13 \right) } = \color{orangered}{ -21 } $
$$ \begin{array}{c|rrrrr}-1&-11&2&\color{orangered}{ -8 }&0&-4\\& & 11& \color{orangered}{-13} & & \\ \hline &-11&13&\color{orangered}{-21}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -21 \right) } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&-11&2&-8&0&-4\\& & 11& -13& \color{blue}{21} & \\ \hline &-11&13&\color{blue}{-21}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 21 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrrr}-1&-11&2&-8&\color{orangered}{ 0 }&-4\\& & 11& -13& \color{orangered}{21} & \\ \hline &-11&13&-21&\color{orangered}{21}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 21 } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&-11&2&-8&0&-4\\& & 11& -13& 21& \color{blue}{-21} \\ \hline &-11&13&-21&\color{blue}{21}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ -25 } $
$$ \begin{array}{c|rrrrr}-1&-11&2&-8&0&\color{orangered}{ -4 }\\& & 11& -13& 21& \color{orangered}{-21} \\ \hline &\color{blue}{-11}&\color{blue}{13}&\color{blue}{-21}&\color{blue}{21}&\color{orangered}{-25} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -11x^{3}+13x^{2}-21x+21 } $ with a remainder of $ \color{red}{ -25 } $.