Divide using synthetic division $$ ( 2x^{5}-6x^{4}-37x^{3}-28x^{2}+8x-8 ) \div ( x+2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-2&2&-6&-37&-28&8&-8\\& & -4& 20& 34& -12& \color{black}{8} \\ \hline &\color{blue}{2}&\color{blue}{-10}&\color{blue}{-17}&\color{blue}{6}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ 2x^{5}-6x^{4}-37x^{3}-28x^{2}+8x-8 }{ x+2 } = \color{blue}{2x^{4}-10x^{3}-17x^{2}+6x-4} $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&-6&-37&-28&8&-8\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-2&\color{orangered}{ 2 }&-6&-37&-28&8&-8\\& & & & & & \\ \hline &\color{orangered}{2}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&-6&-37&-28&8&-8\\& & \color{blue}{-4} & & & & \\ \hline &\color{blue}{2}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrrr}-2&2&\color{orangered}{ -6 }&-37&-28&8&-8\\& & \color{orangered}{-4} & & & & \\ \hline &2&\color{orangered}{-10}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&-6&-37&-28&8&-8\\& & -4& \color{blue}{20} & & & \\ \hline &2&\color{blue}{-10}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -37 } + \color{orangered}{ 20 } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrrr}-2&2&-6&\color{orangered}{ -37 }&-28&8&-8\\& & -4& \color{orangered}{20} & & & \\ \hline &2&-10&\color{orangered}{-17}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 34 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&-6&-37&-28&8&-8\\& & -4& 20& \color{blue}{34} & & \\ \hline &2&-10&\color{blue}{-17}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 34 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrr}-2&2&-6&-37&\color{orangered}{ -28 }&8&-8\\& & -4& 20& \color{orangered}{34} & & \\ \hline &2&-10&-17&\color{orangered}{6}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 6 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&-6&-37&-28&8&-8\\& & -4& 20& 34& \color{blue}{-12} & \\ \hline &2&-10&-17&\color{blue}{6}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}-2&2&-6&-37&-28&\color{orangered}{ 8 }&-8\\& & -4& 20& 34& \color{orangered}{-12} & \\ \hline &2&-10&-17&6&\color{orangered}{-4}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&-6&-37&-28&8&-8\\& & -4& 20& 34& -12& \color{blue}{8} \\ \hline &2&-10&-17&6&\color{blue}{-4}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 8 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-2&2&-6&-37&-28&8&\color{orangered}{ -8 }\\& & -4& 20& 34& -12& \color{orangered}{8} \\ \hline &\color{blue}{2}&\color{blue}{-10}&\color{blue}{-17}&\color{blue}{6}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{4}-10x^{3}-17x^{2}+6x-4 } $ with a remainder of $ \color{red}{ 0 } $.