Divide using synthetic division $$ ( 2x^{5}+3x^{4}-8x^{2}-5x+2 ) \div ( x+2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-2&2&3&0&-8&-5&2\\& & -4& 2& -4& 24& \color{black}{-38} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{2}&\color{blue}{-12}&\color{blue}{19}&\color{orangered}{-36} \end{array} $$The solution is:
$$ \dfrac{ 2x^{5}+3x^{4}-8x^{2}-5x+2 }{ x+2 } = \color{blue}{2x^{4}-x^{3}+2x^{2}-12x+19} \color{red}{~-~} \dfrac{ \color{red}{ 36 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&3&0&-8&-5&2\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-2&\color{orangered}{ 2 }&3&0&-8&-5&2\\& & & & & & \\ \hline &\color{orangered}{2}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&3&0&-8&-5&2\\& & \color{blue}{-4} & & & & \\ \hline &\color{blue}{2}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrr}-2&2&\color{orangered}{ 3 }&0&-8&-5&2\\& & \color{orangered}{-4} & & & & \\ \hline &2&\color{orangered}{-1}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&3&0&-8&-5&2\\& & -4& \color{blue}{2} & & & \\ \hline &2&\color{blue}{-1}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrr}-2&2&3&\color{orangered}{ 0 }&-8&-5&2\\& & -4& \color{orangered}{2} & & & \\ \hline &2&-1&\color{orangered}{2}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&3&0&-8&-5&2\\& & -4& 2& \color{blue}{-4} & & \\ \hline &2&-1&\color{blue}{2}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrrr}-2&2&3&0&\color{orangered}{ -8 }&-5&2\\& & -4& 2& \color{orangered}{-4} & & \\ \hline &2&-1&2&\color{orangered}{-12}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&3&0&-8&-5&2\\& & -4& 2& -4& \color{blue}{24} & \\ \hline &2&-1&2&\color{blue}{-12}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 24 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrrr}-2&2&3&0&-8&\color{orangered}{ -5 }&2\\& & -4& 2& -4& \color{orangered}{24} & \\ \hline &2&-1&2&-12&\color{orangered}{19}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 19 } = \color{blue}{ -38 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&3&0&-8&-5&2\\& & -4& 2& -4& 24& \color{blue}{-38} \\ \hline &2&-1&2&-12&\color{blue}{19}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -38 \right) } = \color{orangered}{ -36 } $
$$ \begin{array}{c|rrrrrr}-2&2&3&0&-8&-5&\color{orangered}{ 2 }\\& & -4& 2& -4& 24& \color{orangered}{-38} \\ \hline &\color{blue}{2}&\color{blue}{-1}&\color{blue}{2}&\color{blue}{-12}&\color{blue}{19}&\color{orangered}{-36} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{4}-x^{3}+2x^{2}-12x+19 } $ with a remainder of $ \color{red}{ -36 } $.