Divide using synthetic division $$ ( 2x^{5}-4x^{3}+4x^{2}-4 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-5&2&0&-4&4&0&-4\\& & -10& 50& -230& 1130& \color{black}{-5650} \\ \hline &\color{blue}{2}&\color{blue}{-10}&\color{blue}{46}&\color{blue}{-226}&\color{blue}{1130}&\color{orangered}{-5654} \end{array} $$The solution is:
$$ \frac{ 2x^{5}-4x^{3}+4x^{2}-4 }{ x+5 } = \color{blue}{2x^{4}-10x^{3}+46x^{2}-226x+1130} \color{red}{~-~} \frac{ \color{red}{ 5654 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&2&0&-4&4&0&-4\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-5&\color{orangered}{ 2 }&0&-4&4&0&-4\\& & & & & & \\ \hline &\color{orangered}{2}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&2&0&-4&4&0&-4\\& & \color{blue}{-10} & & & & \\ \hline &\color{blue}{2}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrrr}-5&2&\color{orangered}{ 0 }&-4&4&0&-4\\& & \color{orangered}{-10} & & & & \\ \hline &2&\color{orangered}{-10}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 50 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&2&0&-4&4&0&-4\\& & -10& \color{blue}{50} & & & \\ \hline &2&\color{blue}{-10}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 50 } = \color{orangered}{ 46 } $
$$ \begin{array}{c|rrrrrr}-5&2&0&\color{orangered}{ -4 }&4&0&-4\\& & -10& \color{orangered}{50} & & & \\ \hline &2&-10&\color{orangered}{46}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 46 } = \color{blue}{ -230 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&2&0&-4&4&0&-4\\& & -10& 50& \color{blue}{-230} & & \\ \hline &2&-10&\color{blue}{46}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -230 \right) } = \color{orangered}{ -226 } $
$$ \begin{array}{c|rrrrrr}-5&2&0&-4&\color{orangered}{ 4 }&0&-4\\& & -10& 50& \color{orangered}{-230} & & \\ \hline &2&-10&46&\color{orangered}{-226}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -226 \right) } = \color{blue}{ 1130 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&2&0&-4&4&0&-4\\& & -10& 50& -230& \color{blue}{1130} & \\ \hline &2&-10&46&\color{blue}{-226}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 1130 } = \color{orangered}{ 1130 } $
$$ \begin{array}{c|rrrrrr}-5&2&0&-4&4&\color{orangered}{ 0 }&-4\\& & -10& 50& -230& \color{orangered}{1130} & \\ \hline &2&-10&46&-226&\color{orangered}{1130}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1130 } = \color{blue}{ -5650 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&2&0&-4&4&0&-4\\& & -10& 50& -230& 1130& \color{blue}{-5650} \\ \hline &2&-10&46&-226&\color{blue}{1130}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -5650 \right) } = \color{orangered}{ -5654 } $
$$ \begin{array}{c|rrrrrr}-5&2&0&-4&4&0&\color{orangered}{ -4 }\\& & -10& 50& -230& 1130& \color{orangered}{-5650} \\ \hline &\color{blue}{2}&\color{blue}{-10}&\color{blue}{46}&\color{blue}{-226}&\color{blue}{1130}&\color{orangered}{-5654} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{4}-10x^{3}+46x^{2}-226x+1130 } $ with a remainder of $ \color{red}{ -5654 } $.