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