Divide using synthetic division $$ ( -2x^{4}+4x^{2}+6x+4 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&-2&0&4&6&4\\& & 4& -8& 8& \color{black}{-28} \\ \hline &\color{blue}{-2}&\color{blue}{4}&\color{blue}{-4}&\color{blue}{14}&\color{orangered}{-24} \end{array} $$The solution is:
$$ \frac{ -2x^{4}+4x^{2}+6x+4 }{ x+2 } = \color{blue}{-2x^{3}+4x^{2}-4x+14} \color{red}{~-~} \frac{ \color{red}{ 24 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-2&0&4&6&4\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ -2 }&0&4&6&4\\& & & & & \\ \hline &\color{orangered}{-2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-2&0&4&6&4\\& & \color{blue}{4} & & & \\ \hline &\color{blue}{-2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}-2&-2&\color{orangered}{ 0 }&4&6&4\\& & \color{orangered}{4} & & & \\ \hline &-2&\color{orangered}{4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-2&0&4&6&4\\& & 4& \color{blue}{-8} & & \\ \hline &-2&\color{blue}{4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}-2&-2&0&\color{orangered}{ 4 }&6&4\\& & 4& \color{orangered}{-8} & & \\ \hline &-2&4&\color{orangered}{-4}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-2&0&4&6&4\\& & 4& -8& \color{blue}{8} & \\ \hline &-2&4&\color{blue}{-4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 8 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrrr}-2&-2&0&4&\color{orangered}{ 6 }&4\\& & 4& -8& \color{orangered}{8} & \\ \hline &-2&4&-4&\color{orangered}{14}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 14 } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-2&0&4&6&4\\& & 4& -8& 8& \color{blue}{-28} \\ \hline &-2&4&-4&\color{blue}{14}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrrr}-2&-2&0&4&6&\color{orangered}{ 4 }\\& & 4& -8& 8& \color{orangered}{-28} \\ \hline &\color{blue}{-2}&\color{blue}{4}&\color{blue}{-4}&\color{blue}{14}&\color{orangered}{-24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{3}+4x^{2}-4x+14 } $ with a remainder of $ \color{red}{ -24 } $.