Divide using synthetic division $$ ( 8x^{3}-6x^{2}+4x-2 ) \div ( 2x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&8&-6&4&-2\\& & -32& 152& \color{black}{-624} \\ \hline &\color{blue}{8}&\color{blue}{-38}&\color{blue}{156}&\color{orangered}{-626} \end{array} $$The solution is:
$$ \frac{ 8x^{3}-6x^{2}+4x-2 }{ x+4 } = \color{blue}{8x^{2}-38x+156} \color{red}{~-~} \frac{ \color{red}{ 626 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&8&-6&4&-2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 8 }&-6&4&-2\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 8 } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&8&-6&4&-2\\& & \color{blue}{-32} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -38 } $
$$ \begin{array}{c|rrrr}-4&8&\color{orangered}{ -6 }&4&-2\\& & \color{orangered}{-32} & & \\ \hline &8&\color{orangered}{-38}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -38 \right) } = \color{blue}{ 152 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&8&-6&4&-2\\& & -32& \color{blue}{152} & \\ \hline &8&\color{blue}{-38}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 152 } = \color{orangered}{ 156 } $
$$ \begin{array}{c|rrrr}-4&8&-6&\color{orangered}{ 4 }&-2\\& & -32& \color{orangered}{152} & \\ \hline &8&-38&\color{orangered}{156}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 156 } = \color{blue}{ -624 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&8&-6&4&-2\\& & -32& 152& \color{blue}{-624} \\ \hline &8&-38&\color{blue}{156}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -624 \right) } = \color{orangered}{ -626 } $
$$ \begin{array}{c|rrrr}-4&8&-6&4&\color{orangered}{ -2 }\\& & -32& 152& \color{orangered}{-624} \\ \hline &\color{blue}{8}&\color{blue}{-38}&\color{blue}{156}&\color{orangered}{-626} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}-38x+156 } $ with a remainder of $ \color{red}{ -626 } $.