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