Divide using synthetic division $$ ( 202x^{4}+7x^{3}-4x^{2}-10x+20 ) \div ( -2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&202&7&-4&-10&20\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{202}&\color{blue}{7}&\color{blue}{-4}&\color{blue}{-10}&\color{orangered}{20} \end{array} $$The solution is:
$$ \frac{ 202x^{4}+7x^{3}-4x^{2}-10x+20 }{ x } = \color{blue}{202x^{3}+7x^{2}-4x-10} ~+~ \frac{ \color{red}{ 20 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&202&7&-4&-10&20\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 202 }&7&-4&-10&20\\& & & & & \\ \hline &\color{orangered}{202}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 202 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&202&7&-4&-10&20\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{202}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 0 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrr}0&202&\color{orangered}{ 7 }&-4&-10&20\\& & \color{orangered}{0} & & & \\ \hline &202&\color{orangered}{7}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 7 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&202&7&-4&-10&20\\& & 0& \color{blue}{0} & & \\ \hline &202&\color{blue}{7}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 0 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}0&202&7&\color{orangered}{ -4 }&-10&20\\& & 0& \color{orangered}{0} & & \\ \hline &202&7&\color{orangered}{-4}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&202&7&-4&-10&20\\& & 0& 0& \color{blue}{0} & \\ \hline &202&7&\color{blue}{-4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 0 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrr}0&202&7&-4&\color{orangered}{ -10 }&20\\& & 0& 0& \color{orangered}{0} & \\ \hline &202&7&-4&\color{orangered}{-10}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&202&7&-4&-10&20\\& & 0& 0& 0& \color{blue}{0} \\ \hline &202&7&-4&\color{blue}{-10}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ 0 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrrrr}0&202&7&-4&-10&\color{orangered}{ 20 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{202}&\color{blue}{7}&\color{blue}{-4}&\color{blue}{-10}&\color{orangered}{20} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 202x^{3}+7x^{2}-4x-10 } $ with a remainder of $ \color{red}{ 20 } $.