Divide using synthetic division $$ ( -x^{4}-5x^{3}+3x+1 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&-1&-5&0&3&1\\& & 2& 6& -12& \color{black}{18} \\ \hline &\color{blue}{-1}&\color{blue}{-3}&\color{blue}{6}&\color{blue}{-9}&\color{orangered}{19} \end{array} $$The solution is:
$$ \frac{ -x^{4}-5x^{3}+3x+1 }{ x+2 } = \color{blue}{-x^{3}-3x^{2}+6x-9} ~+~ \frac{ \color{red}{ 19 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-1&-5&0&3&1\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ -1 }&-5&0&3&1\\& & & & & \\ \hline &\color{orangered}{-1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-1&-5&0&3&1\\& & \color{blue}{2} & & & \\ \hline &\color{blue}{-1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 2 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}-2&-1&\color{orangered}{ -5 }&0&3&1\\& & \color{orangered}{2} & & & \\ \hline &-1&\color{orangered}{-3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-1&-5&0&3&1\\& & 2& \color{blue}{6} & & \\ \hline &-1&\color{blue}{-3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 6 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}-2&-1&-5&\color{orangered}{ 0 }&3&1\\& & 2& \color{orangered}{6} & & \\ \hline &-1&-3&\color{orangered}{6}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 6 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-1&-5&0&3&1\\& & 2& 6& \color{blue}{-12} & \\ \hline &-1&-3&\color{blue}{6}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-2&-1&-5&0&\color{orangered}{ 3 }&1\\& & 2& 6& \color{orangered}{-12} & \\ \hline &-1&-3&6&\color{orangered}{-9}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-1&-5&0&3&1\\& & 2& 6& -12& \color{blue}{18} \\ \hline &-1&-3&6&\color{blue}{-9}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 18 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrr}-2&-1&-5&0&3&\color{orangered}{ 1 }\\& & 2& 6& -12& \color{orangered}{18} \\ \hline &\color{blue}{-1}&\color{blue}{-3}&\color{blue}{6}&\color{blue}{-9}&\color{orangered}{19} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{3}-3x^{2}+6x-9 } $ with a remainder of $ \color{red}{ 19 } $.