Divide using synthetic division $$ ( -4x^{5}-5x^{4}+10x^{3}+3x^{2}+10x+9 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-1&-4&-5&10&3&10&9\\& & 4& 1& -11& 8& \color{black}{-18} \\ \hline &\color{blue}{-4}&\color{blue}{-1}&\color{blue}{11}&\color{blue}{-8}&\color{blue}{18}&\color{orangered}{-9} \end{array} $$The solution is:
$$ \frac{ -4x^{5}-5x^{4}+10x^{3}+3x^{2}+10x+9 }{ x+1 } = \color{blue}{-4x^{4}-x^{3}+11x^{2}-8x+18} \color{red}{~-~} \frac{ \color{red}{ 9 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-4&-5&10&3&10&9\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-1&\color{orangered}{ -4 }&-5&10&3&10&9\\& & & & & & \\ \hline &\color{orangered}{-4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-4&-5&10&3&10&9\\& & \color{blue}{4} & & & & \\ \hline &\color{blue}{-4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 4 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrr}-1&-4&\color{orangered}{ -5 }&10&3&10&9\\& & \color{orangered}{4} & & & & \\ \hline &-4&\color{orangered}{-1}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-4&-5&10&3&10&9\\& & 4& \color{blue}{1} & & & \\ \hline &-4&\color{blue}{-1}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 1 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrrr}-1&-4&-5&\color{orangered}{ 10 }&3&10&9\\& & 4& \color{orangered}{1} & & & \\ \hline &-4&-1&\color{orangered}{11}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 11 } = \color{blue}{ -11 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-4&-5&10&3&10&9\\& & 4& 1& \color{blue}{-11} & & \\ \hline &-4&-1&\color{blue}{11}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -11 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrrr}-1&-4&-5&10&\color{orangered}{ 3 }&10&9\\& & 4& 1& \color{orangered}{-11} & & \\ \hline &-4&-1&11&\color{orangered}{-8}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-4&-5&10&3&10&9\\& & 4& 1& -11& \color{blue}{8} & \\ \hline &-4&-1&11&\color{blue}{-8}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 8 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrr}-1&-4&-5&10&3&\color{orangered}{ 10 }&9\\& & 4& 1& -11& \color{orangered}{8} & \\ \hline &-4&-1&11&-8&\color{orangered}{18}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 18 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-1}&-4&-5&10&3&10&9\\& & 4& 1& -11& 8& \color{blue}{-18} \\ \hline &-4&-1&11&-8&\color{blue}{18}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrrr}-1&-4&-5&10&3&10&\color{orangered}{ 9 }\\& & 4& 1& -11& 8& \color{orangered}{-18} \\ \hline &\color{blue}{-4}&\color{blue}{-1}&\color{blue}{11}&\color{blue}{-8}&\color{blue}{18}&\color{orangered}{-9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x^{4}-x^{3}+11x^{2}-8x+18 } $ with a remainder of $ \color{red}{ -9 } $.