Divide using synthetic division $$ ( -x^{5}+5x^{4}+4x^{3}-20x^{2} ) \div ( 4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}0&-1&5&4&-20&0&0\\& & 0& 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{-1}&\color{blue}{5}&\color{blue}{4}&\color{blue}{-20}&\color{blue}{0}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -x^{5}+5x^{4}+4x^{3}-20x^{2} }{ x } = \color{blue}{-x^{4}+5x^{3}+4x^{2}-20x} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&-1&5&4&-20&0&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}0&\color{orangered}{ -1 }&5&4&-20&0&0\\& & & & & & \\ \hline &\color{orangered}{-1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&-1&5&4&-20&0&0\\& & \color{blue}{0} & & & & \\ \hline &\color{blue}{-1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 0 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrr}0&-1&\color{orangered}{ 5 }&4&-20&0&0\\& & \color{orangered}{0} & & & & \\ \hline &-1&\color{orangered}{5}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 5 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&-1&5&4&-20&0&0\\& & 0& \color{blue}{0} & & & \\ \hline &-1&\color{blue}{5}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 0 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}0&-1&5&\color{orangered}{ 4 }&-20&0&0\\& & 0& \color{orangered}{0} & & & \\ \hline &-1&5&\color{orangered}{4}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 4 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&-1&5&4&-20&0&0\\& & 0& 0& \color{blue}{0} & & \\ \hline &-1&5&\color{blue}{4}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 0 } = \color{orangered}{ -20 } $
$$ \begin{array}{c|rrrrrr}0&-1&5&4&\color{orangered}{ -20 }&0&0\\& & 0& 0& \color{orangered}{0} & & \\ \hline &-1&5&4&\color{orangered}{-20}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -20 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&-1&5&4&-20&0&0\\& & 0& 0& 0& \color{blue}{0} & \\ \hline &-1&5&4&\color{blue}{-20}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}0&-1&5&4&-20&\color{orangered}{ 0 }&0\\& & 0& 0& 0& \color{orangered}{0} & \\ \hline &-1&5&4&-20&\color{orangered}{0}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&-1&5&4&-20&0&0\\& & 0& 0& 0& 0& \color{blue}{0} \\ \hline &-1&5&4&-20&\color{blue}{0}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}0&-1&5&4&-20&0&\color{orangered}{ 0 }\\& & 0& 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{-1}&\color{blue}{5}&\color{blue}{4}&\color{blue}{-20}&\color{blue}{0}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{4}+5x^{3}+4x^{2}-20x } $ with a remainder of $ \color{red}{ 0 } $.