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