Divide using synthetic division $$ ( x^{7}+2x^{6}-3x^{5}+2x+2 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrr}-1&1&2&-3&0&0&0&2&2\\& & -1& -1& 4& -4& 4& -4& \color{black}{2} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{-4}&\color{blue}{4}&\color{blue}{-4}&\color{blue}{4}&\color{blue}{-2}&\color{orangered}{4} \end{array} $$The solution is:
$$ \frac{ x^{7}+2x^{6}-3x^{5}+2x+2 }{ x+1 } = \color{blue}{x^{6}+x^{5}-4x^{4}+4x^{3}-4x^{2}+4x-2} ~+~ \frac{ \color{red}{ 4 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&1&2&-3&0&0&0&2&2\\& & & & & & & & \\ \hline &&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrr}-1&\color{orangered}{ 1 }&2&-3&0&0&0&2&2\\& & & & & & & & \\ \hline &\color{orangered}{1}&&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 1 } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&1&2&-3&0&0&0&2&2\\& & \color{blue}{-1} & & & & & & \\ \hline &\color{blue}{1}&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrrrr}-1&1&\color{orangered}{ 2 }&-3&0&0&0&2&2\\& & \color{orangered}{-1} & & & & & & \\ \hline &1&\color{orangered}{1}&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 1 } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&1&2&-3&0&0&0&2&2\\& & -1& \color{blue}{-1} & & & & & \\ \hline &1&\color{blue}{1}&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrrrr}-1&1&2&\color{orangered}{ -3 }&0&0&0&2&2\\& & -1& \color{orangered}{-1} & & & & & \\ \hline &1&1&\color{orangered}{-4}&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&1&2&-3&0&0&0&2&2\\& & -1& -1& \color{blue}{4} & & & & \\ \hline &1&1&\color{blue}{-4}&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrrrr}-1&1&2&-3&\color{orangered}{ 0 }&0&0&2&2\\& & -1& -1& \color{orangered}{4} & & & & \\ \hline &1&1&-4&\color{orangered}{4}&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 4 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&1&2&-3&0&0&0&2&2\\& & -1& -1& 4& \color{blue}{-4} & & & \\ \hline &1&1&-4&\color{blue}{4}&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrrrr}-1&1&2&-3&0&\color{orangered}{ 0 }&0&2&2\\& & -1& -1& 4& \color{orangered}{-4} & & & \\ \hline &1&1&-4&4&\color{orangered}{-4}&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&1&2&-3&0&0&0&2&2\\& & -1& -1& 4& -4& \color{blue}{4} & & \\ \hline &1&1&-4&4&\color{blue}{-4}&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrrrr}-1&1&2&-3&0&0&\color{orangered}{ 0 }&2&2\\& & -1& -1& 4& -4& \color{orangered}{4} & & \\ \hline &1&1&-4&4&-4&\color{orangered}{4}&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 4 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&1&2&-3&0&0&0&2&2\\& & -1& -1& 4& -4& 4& \color{blue}{-4} & \\ \hline &1&1&-4&4&-4&\color{blue}{4}&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrrrr}-1&1&2&-3&0&0&0&\color{orangered}{ 2 }&2\\& & -1& -1& 4& -4& 4& \color{orangered}{-4} & \\ \hline &1&1&-4&4&-4&4&\color{orangered}{-2}& \end{array} $$Step 14 : 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 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-1}&1&2&-3&0&0&0&2&2\\& & -1& -1& 4& -4& 4& -4& \color{blue}{2} \\ \hline &1&1&-4&4&-4&4&\color{blue}{-2}& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 2 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrrrr}-1&1&2&-3&0&0&0&2&\color{orangered}{ 2 }\\& & -1& -1& 4& -4& 4& -4& \color{orangered}{2} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{-4}&\color{blue}{4}&\color{blue}{-4}&\color{blue}{4}&\color{blue}{-2}&\color{orangered}{4} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{6}+x^{5}-4x^{4}+4x^{3}-4x^{2}+4x-2 } $ with a remainder of $ \color{red}{ 4 } $.