Divide using synthetic division $$ ( x^{6}+3x^{3}-3x^{2}-3x+2 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-2&1&0&0&3&-3&-3&2\\& & -2& 4& -8& 10& -14& \color{black}{34} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{4}&\color{blue}{-5}&\color{blue}{7}&\color{blue}{-17}&\color{orangered}{36} \end{array} $$The solution is:
$$ \frac{ x^{6}+3x^{3}-3x^{2}-3x+2 }{ x+2 } = \color{blue}{x^{5}-2x^{4}+4x^{3}-5x^{2}+7x-17} ~+~ \frac{ \color{red}{ 36 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&0&3&-3&-3&2\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-2&\color{orangered}{ 1 }&0&0&3&-3&-3&2\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&0&3&-3&-3&2\\& & \color{blue}{-2} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrrr}-2&1&\color{orangered}{ 0 }&0&3&-3&-3&2\\& & \color{orangered}{-2} & & & & & \\ \hline &1&\color{orangered}{-2}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&0&3&-3&-3&2\\& & -2& \color{blue}{4} & & & & \\ \hline &1&\color{blue}{-2}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&\color{orangered}{ 0 }&3&-3&-3&2\\& & -2& \color{orangered}{4} & & & & \\ \hline &1&-2&\color{orangered}{4}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&0&3&-3&-3&2\\& & -2& 4& \color{blue}{-8} & & & \\ \hline &1&-2&\color{blue}{4}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&0&\color{orangered}{ 3 }&-3&-3&2\\& & -2& 4& \color{orangered}{-8} & & & \\ \hline &1&-2&4&\color{orangered}{-5}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&0&3&-3&-3&2\\& & -2& 4& -8& \color{blue}{10} & & \\ \hline &1&-2&4&\color{blue}{-5}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 10 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&0&3&\color{orangered}{ -3 }&-3&2\\& & -2& 4& -8& \color{orangered}{10} & & \\ \hline &1&-2&4&-5&\color{orangered}{7}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 7 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&0&3&-3&-3&2\\& & -2& 4& -8& 10& \color{blue}{-14} & \\ \hline &1&-2&4&-5&\color{blue}{7}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&0&3&-3&\color{orangered}{ -3 }&2\\& & -2& 4& -8& 10& \color{orangered}{-14} & \\ \hline &1&-2&4&-5&7&\color{orangered}{-17}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 34 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&0&3&-3&-3&2\\& & -2& 4& -8& 10& -14& \color{blue}{34} \\ \hline &1&-2&4&-5&7&\color{blue}{-17}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 34 } = \color{orangered}{ 36 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&0&3&-3&-3&\color{orangered}{ 2 }\\& & -2& 4& -8& 10& -14& \color{orangered}{34} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{4}&\color{blue}{-5}&\color{blue}{7}&\color{blue}{-17}&\color{orangered}{36} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{5}-2x^{4}+4x^{3}-5x^{2}+7x-17 } $ with a remainder of $ \color{red}{ 36 } $.