Divide using synthetic division $$ ( x^{5}+3x^{4}-2x^{3}+2x-10 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-2&1&3&-2&0&2&-10\\& & -2& -2& 8& -16& \color{black}{28} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{-4}&\color{blue}{8}&\color{blue}{-14}&\color{orangered}{18} \end{array} $$The solution is:
$$ \frac{ x^{5}+3x^{4}-2x^{3}+2x-10 }{ x+2 } = \color{blue}{x^{4}+x^{3}-4x^{2}+8x-14} ~+~ \frac{ \color{red}{ 18 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&3&-2&0&2&-10\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-2&\color{orangered}{ 1 }&3&-2&0&2&-10\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&3&-2&0&2&-10\\& & \color{blue}{-2} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrr}-2&1&\color{orangered}{ 3 }&-2&0&2&-10\\& & \color{orangered}{-2} & & & & \\ \hline &1&\color{orangered}{1}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&3&-2&0&2&-10\\& & -2& \color{blue}{-2} & & & \\ \hline &1&\color{blue}{1}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}-2&1&3&\color{orangered}{ -2 }&0&2&-10\\& & -2& \color{orangered}{-2} & & & \\ \hline &1&1&\color{orangered}{-4}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&3&-2&0&2&-10\\& & -2& -2& \color{blue}{8} & & \\ \hline &1&1&\color{blue}{-4}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrr}-2&1&3&-2&\color{orangered}{ 0 }&2&-10\\& & -2& -2& \color{orangered}{8} & & \\ \hline &1&1&-4&\color{orangered}{8}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 8 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&3&-2&0&2&-10\\& & -2& -2& 8& \color{blue}{-16} & \\ \hline &1&1&-4&\color{blue}{8}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrrrrr}-2&1&3&-2&0&\color{orangered}{ 2 }&-10\\& & -2& -2& 8& \color{orangered}{-16} & \\ \hline &1&1&-4&8&\color{orangered}{-14}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&3&-2&0&2&-10\\& & -2& -2& 8& -16& \color{blue}{28} \\ \hline &1&1&-4&8&\color{blue}{-14}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 28 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrr}-2&1&3&-2&0&2&\color{orangered}{ -10 }\\& & -2& -2& 8& -16& \color{orangered}{28} \\ \hline &\color{blue}{1}&\color{blue}{1}&\color{blue}{-4}&\color{blue}{8}&\color{blue}{-14}&\color{orangered}{18} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+x^{3}-4x^{2}+8x-14 } $ with a remainder of $ \color{red}{ 18 } $.