Divide using synthetic division $$ ( 8x^{5}-x^{4}+2x^{2}-6x+2 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-2&8&-1&0&2&-6&2\\& & -16& 34& -68& 132& \color{black}{-252} \\ \hline &\color{blue}{8}&\color{blue}{-17}&\color{blue}{34}&\color{blue}{-66}&\color{blue}{126}&\color{orangered}{-250} \end{array} $$The solution is:
$$ \frac{ 8x^{5}-x^{4}+2x^{2}-6x+2 }{ x+2 } = \color{blue}{8x^{4}-17x^{3}+34x^{2}-66x+126} \color{red}{~-~} \frac{ \color{red}{ 250 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&8&-1&0&2&-6&2\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-2&\color{orangered}{ 8 }&-1&0&2&-6&2\\& & & & & & \\ \hline &\color{orangered}{8}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 8 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&8&-1&0&2&-6&2\\& & \color{blue}{-16} & & & & \\ \hline &\color{blue}{8}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrrr}-2&8&\color{orangered}{ -1 }&0&2&-6&2\\& & \color{orangered}{-16} & & & & \\ \hline &8&\color{orangered}{-17}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 34 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&8&-1&0&2&-6&2\\& & -16& \color{blue}{34} & & & \\ \hline &8&\color{blue}{-17}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 34 } = \color{orangered}{ 34 } $
$$ \begin{array}{c|rrrrrr}-2&8&-1&\color{orangered}{ 0 }&2&-6&2\\& & -16& \color{orangered}{34} & & & \\ \hline &8&-17&\color{orangered}{34}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 34 } = \color{blue}{ -68 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&8&-1&0&2&-6&2\\& & -16& 34& \color{blue}{-68} & & \\ \hline &8&-17&\color{blue}{34}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -68 \right) } = \color{orangered}{ -66 } $
$$ \begin{array}{c|rrrrrr}-2&8&-1&0&\color{orangered}{ 2 }&-6&2\\& & -16& 34& \color{orangered}{-68} & & \\ \hline &8&-17&34&\color{orangered}{-66}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -66 \right) } = \color{blue}{ 132 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&8&-1&0&2&-6&2\\& & -16& 34& -68& \color{blue}{132} & \\ \hline &8&-17&34&\color{blue}{-66}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 132 } = \color{orangered}{ 126 } $
$$ \begin{array}{c|rrrrrr}-2&8&-1&0&2&\color{orangered}{ -6 }&2\\& & -16& 34& -68& \color{orangered}{132} & \\ \hline &8&-17&34&-66&\color{orangered}{126}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 126 } = \color{blue}{ -252 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&8&-1&0&2&-6&2\\& & -16& 34& -68& 132& \color{blue}{-252} \\ \hline &8&-17&34&-66&\color{blue}{126}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -252 \right) } = \color{orangered}{ -250 } $
$$ \begin{array}{c|rrrrrr}-2&8&-1&0&2&-6&\color{orangered}{ 2 }\\& & -16& 34& -68& 132& \color{orangered}{-252} \\ \hline &\color{blue}{8}&\color{blue}{-17}&\color{blue}{34}&\color{blue}{-66}&\color{blue}{126}&\color{orangered}{-250} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{4}-17x^{3}+34x^{2}-66x+126 } $ with a remainder of $ \color{red}{ -250 } $.