Divide using synthetic division $$ ( x^{5}-6x^{2}+11x ) \div ( x+2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-2&1&0&0&-6&11&0\\& & -2& 4& -8& 28& \color{black}{-78} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{4}&\color{blue}{-14}&\color{blue}{39}&\color{orangered}{-78} \end{array} $$The solution is:
$$ \dfrac{ x^{5}-6x^{2}+11x }{ x+2 } = \color{blue}{x^{4}-2x^{3}+4x^{2}-14x+39} \color{red}{~-~} \dfrac{ \color{red}{ 78 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&0&0&-6&11&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-2&\color{orangered}{ 1 }&0&0&-6&11&0\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&0&0&-6&11&0\\& & \color{blue}{-2} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrr}-2&1&\color{orangered}{ 0 }&0&-6&11&0\\& & \color{orangered}{-2} & & & & \\ \hline &1&\color{orangered}{-2}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&0&0&-6&11&0\\& & -2& \color{blue}{4} & & & \\ \hline &1&\color{blue}{-2}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}-2&1&0&\color{orangered}{ 0 }&-6&11&0\\& & -2& \color{orangered}{4} & & & \\ \hline &1&-2&\color{orangered}{4}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&0&0&-6&11&0\\& & -2& 4& \color{blue}{-8} & & \\ \hline &1&-2&\color{blue}{4}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrrrrr}-2&1&0&0&\color{orangered}{ -6 }&11&0\\& & -2& 4& \color{orangered}{-8} & & \\ \hline &1&-2&4&\color{orangered}{-14}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&0&0&-6&11&0\\& & -2& 4& -8& \color{blue}{28} & \\ \hline &1&-2&4&\color{blue}{-14}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ 28 } = \color{orangered}{ 39 } $
$$ \begin{array}{c|rrrrrr}-2&1&0&0&-6&\color{orangered}{ 11 }&0\\& & -2& 4& -8& \color{orangered}{28} & \\ \hline &1&-2&4&-14&\color{orangered}{39}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 39 } = \color{blue}{ -78 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&1&0&0&-6&11&0\\& & -2& 4& -8& 28& \color{blue}{-78} \\ \hline &1&-2&4&-14&\color{blue}{39}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -78 \right) } = \color{orangered}{ -78 } $
$$ \begin{array}{c|rrrrrr}-2&1&0&0&-6&11&\color{orangered}{ 0 }\\& & -2& 4& -8& 28& \color{orangered}{-78} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{4}&\color{blue}{-14}&\color{blue}{39}&\color{orangered}{-78} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-2x^{3}+4x^{2}-14x+39 } $ with a remainder of $ \color{red}{ -78 } $.