Divide using synthetic division $$ ( -2x^{4}-4x^{3}-4x^{2}-2 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&-2&-4&-4&0&-2\\& & 10& -30& 170& \color{black}{-850} \\ \hline &\color{blue}{-2}&\color{blue}{6}&\color{blue}{-34}&\color{blue}{170}&\color{orangered}{-852} \end{array} $$The solution is:
$$ \frac{ -2x^{4}-4x^{3}-4x^{2}-2 }{ x+5 } = \color{blue}{-2x^{3}+6x^{2}-34x+170} \color{red}{~-~} \frac{ \color{red}{ 852 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&-2&-4&-4&0&-2\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ -2 }&-4&-4&0&-2\\& & & & & \\ \hline &\color{orangered}{-2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&-2&-4&-4&0&-2\\& & \color{blue}{10} & & & \\ \hline &\color{blue}{-2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 10 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}-5&-2&\color{orangered}{ -4 }&-4&0&-2\\& & \color{orangered}{10} & & & \\ \hline &-2&\color{orangered}{6}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 6 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&-2&-4&-4&0&-2\\& & 10& \color{blue}{-30} & & \\ \hline &-2&\color{blue}{6}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -34 } $
$$ \begin{array}{c|rrrrr}-5&-2&-4&\color{orangered}{ -4 }&0&-2\\& & 10& \color{orangered}{-30} & & \\ \hline &-2&6&\color{orangered}{-34}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -34 \right) } = \color{blue}{ 170 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&-2&-4&-4&0&-2\\& & 10& -30& \color{blue}{170} & \\ \hline &-2&6&\color{blue}{-34}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 170 } = \color{orangered}{ 170 } $
$$ \begin{array}{c|rrrrr}-5&-2&-4&-4&\color{orangered}{ 0 }&-2\\& & 10& -30& \color{orangered}{170} & \\ \hline &-2&6&-34&\color{orangered}{170}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 170 } = \color{blue}{ -850 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&-2&-4&-4&0&-2\\& & 10& -30& 170& \color{blue}{-850} \\ \hline &-2&6&-34&\color{blue}{170}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -850 \right) } = \color{orangered}{ -852 } $
$$ \begin{array}{c|rrrrr}-5&-2&-4&-4&0&\color{orangered}{ -2 }\\& & 10& -30& 170& \color{orangered}{-850} \\ \hline &\color{blue}{-2}&\color{blue}{6}&\color{blue}{-34}&\color{blue}{170}&\color{orangered}{-852} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{3}+6x^{2}-34x+170 } $ with a remainder of $ \color{red}{ -852 } $.