Divide using synthetic division $$ ( -3x^{5}+5x^{4}+3x^{3}+6x^{2}+x+8 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-2&-3&5&3&6&1&8\\& & 6& -22& 38& -88& \color{black}{174} \\ \hline &\color{blue}{-3}&\color{blue}{11}&\color{blue}{-19}&\color{blue}{44}&\color{blue}{-87}&\color{orangered}{182} \end{array} $$The solution is:
$$ \frac{ -3x^{5}+5x^{4}+3x^{3}+6x^{2}+x+8 }{ x+2 } = \color{blue}{-3x^{4}+11x^{3}-19x^{2}+44x-87} ~+~ \frac{ \color{red}{ 182 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&-3&5&3&6&1&8\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-2&\color{orangered}{ -3 }&5&3&6&1&8\\& & & & & & \\ \hline &\color{orangered}{-3}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&-3&5&3&6&1&8\\& & \color{blue}{6} & & & & \\ \hline &\color{blue}{-3}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 6 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrrr}-2&-3&\color{orangered}{ 5 }&3&6&1&8\\& & \color{orangered}{6} & & & & \\ \hline &-3&\color{orangered}{11}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 11 } = \color{blue}{ -22 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&-3&5&3&6&1&8\\& & 6& \color{blue}{-22} & & & \\ \hline &-3&\color{blue}{11}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -22 \right) } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrrrrr}-2&-3&5&\color{orangered}{ 3 }&6&1&8\\& & 6& \color{orangered}{-22} & & & \\ \hline &-3&11&\color{orangered}{-19}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ 38 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&-3&5&3&6&1&8\\& & 6& -22& \color{blue}{38} & & \\ \hline &-3&11&\color{blue}{-19}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 38 } = \color{orangered}{ 44 } $
$$ \begin{array}{c|rrrrrr}-2&-3&5&3&\color{orangered}{ 6 }&1&8\\& & 6& -22& \color{orangered}{38} & & \\ \hline &-3&11&-19&\color{orangered}{44}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 44 } = \color{blue}{ -88 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&-3&5&3&6&1&8\\& & 6& -22& 38& \color{blue}{-88} & \\ \hline &-3&11&-19&\color{blue}{44}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -88 \right) } = \color{orangered}{ -87 } $
$$ \begin{array}{c|rrrrrr}-2&-3&5&3&6&\color{orangered}{ 1 }&8\\& & 6& -22& 38& \color{orangered}{-88} & \\ \hline &-3&11&-19&44&\color{orangered}{-87}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -87 \right) } = \color{blue}{ 174 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&-3&5&3&6&1&8\\& & 6& -22& 38& -88& \color{blue}{174} \\ \hline &-3&11&-19&44&\color{blue}{-87}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 174 } = \color{orangered}{ 182 } $
$$ \begin{array}{c|rrrrrr}-2&-3&5&3&6&1&\color{orangered}{ 8 }\\& & 6& -22& 38& -88& \color{orangered}{174} \\ \hline &\color{blue}{-3}&\color{blue}{11}&\color{blue}{-19}&\color{blue}{44}&\color{blue}{-87}&\color{orangered}{182} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{4}+11x^{3}-19x^{2}+44x-87 } $ with a remainder of $ \color{red}{ 182 } $.