Divide using synthetic division $$ ( x^{5}+9x^{4}+21x^{3}-x^{2}-30x ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}2&1&9&21&-1&-30&0\\& & 2& 22& 86& 170& \color{black}{280} \\ \hline &\color{blue}{1}&\color{blue}{11}&\color{blue}{43}&\color{blue}{85}&\color{blue}{140}&\color{orangered}{280} \end{array} $$The solution is:
$$ \frac{ x^{5}+9x^{4}+21x^{3}-x^{2}-30x }{ x-2 } = \color{blue}{x^{4}+11x^{3}+43x^{2}+85x+140} ~+~ \frac{ \color{red}{ 280 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&9&21&-1&-30&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}2&\color{orangered}{ 1 }&9&21&-1&-30&0\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&9&21&-1&-30&0\\& & \color{blue}{2} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 2 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrrr}2&1&\color{orangered}{ 9 }&21&-1&-30&0\\& & \color{orangered}{2} & & & & \\ \hline &1&\color{orangered}{11}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 11 } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&9&21&-1&-30&0\\& & 2& \color{blue}{22} & & & \\ \hline &1&\color{blue}{11}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ 22 } = \color{orangered}{ 43 } $
$$ \begin{array}{c|rrrrrr}2&1&9&\color{orangered}{ 21 }&-1&-30&0\\& & 2& \color{orangered}{22} & & & \\ \hline &1&11&\color{orangered}{43}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 43 } = \color{blue}{ 86 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&9&21&-1&-30&0\\& & 2& 22& \color{blue}{86} & & \\ \hline &1&11&\color{blue}{43}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 86 } = \color{orangered}{ 85 } $
$$ \begin{array}{c|rrrrrr}2&1&9&21&\color{orangered}{ -1 }&-30&0\\& & 2& 22& \color{orangered}{86} & & \\ \hline &1&11&43&\color{orangered}{85}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 85 } = \color{blue}{ 170 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&9&21&-1&-30&0\\& & 2& 22& 86& \color{blue}{170} & \\ \hline &1&11&43&\color{blue}{85}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 170 } = \color{orangered}{ 140 } $
$$ \begin{array}{c|rrrrrr}2&1&9&21&-1&\color{orangered}{ -30 }&0\\& & 2& 22& 86& \color{orangered}{170} & \\ \hline &1&11&43&85&\color{orangered}{140}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 140 } = \color{blue}{ 280 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&9&21&-1&-30&0\\& & 2& 22& 86& 170& \color{blue}{280} \\ \hline &1&11&43&85&\color{blue}{140}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 280 } = \color{orangered}{ 280 } $
$$ \begin{array}{c|rrrrrr}2&1&9&21&-1&-30&\color{orangered}{ 0 }\\& & 2& 22& 86& 170& \color{orangered}{280} \\ \hline &\color{blue}{1}&\color{blue}{11}&\color{blue}{43}&\color{blue}{85}&\color{blue}{140}&\color{orangered}{280} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+11x^{3}+43x^{2}+85x+140 } $ with a remainder of $ \color{red}{ 280 } $.