Divide using synthetic division $$ ( x^{5}+9x^{4}+21x^{3}-x^{2}-30x ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-6&1&9&21&-1&-30&0\\& & -6& -18& -18& 114& \color{black}{-504} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{3}&\color{blue}{-19}&\color{blue}{84}&\color{orangered}{-504} \end{array} $$The solution is:
$$ \frac{ x^{5}+9x^{4}+21x^{3}-x^{2}-30x }{ x+6 } = \color{blue}{x^{4}+3x^{3}+3x^{2}-19x+84} \color{red}{~-~} \frac{ \color{red}{ 504 } }{ x+6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&9&21&-1&-30&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-6&\color{orangered}{ 1 }&9&21&-1&-30&0\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 1 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&9&21&-1&-30&0\\& & \color{blue}{-6} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrr}-6&1&\color{orangered}{ 9 }&21&-1&-30&0\\& & \color{orangered}{-6} & & & & \\ \hline &1&\color{orangered}{3}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 3 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&9&21&-1&-30&0\\& & -6& \color{blue}{-18} & & & \\ \hline &1&\color{blue}{3}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrr}-6&1&9&\color{orangered}{ 21 }&-1&-30&0\\& & -6& \color{orangered}{-18} & & & \\ \hline &1&3&\color{orangered}{3}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 3 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&9&21&-1&-30&0\\& & -6& -18& \color{blue}{-18} & & \\ \hline &1&3&\color{blue}{3}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrrrrr}-6&1&9&21&\color{orangered}{ -1 }&-30&0\\& & -6& -18& \color{orangered}{-18} & & \\ \hline &1&3&3&\color{orangered}{-19}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ 114 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&9&21&-1&-30&0\\& & -6& -18& -18& \color{blue}{114} & \\ \hline &1&3&3&\color{blue}{-19}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 114 } = \color{orangered}{ 84 } $
$$ \begin{array}{c|rrrrrr}-6&1&9&21&-1&\color{orangered}{ -30 }&0\\& & -6& -18& -18& \color{orangered}{114} & \\ \hline &1&3&3&-19&\color{orangered}{84}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 84 } = \color{blue}{ -504 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-6}&1&9&21&-1&-30&0\\& & -6& -18& -18& 114& \color{blue}{-504} \\ \hline &1&3&3&-19&\color{blue}{84}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -504 \right) } = \color{orangered}{ -504 } $
$$ \begin{array}{c|rrrrrr}-6&1&9&21&-1&-30&\color{orangered}{ 0 }\\& & -6& -18& -18& 114& \color{orangered}{-504} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{3}&\color{blue}{-19}&\color{blue}{84}&\color{orangered}{-504} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+3x^{3}+3x^{2}-19x+84 } $ with a remainder of $ \color{red}{ -504 } $.