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