Divide using synthetic division $$ ( 6x^{6}+46x^{5}-69x^{4}+36x^{3}+84x^{2}+35x+71 ) \div ( x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-9&6&46&-69&36&84&35&71\\& & -54& 72& -27& -81& -27& \color{black}{-72} \\ \hline &\color{blue}{6}&\color{blue}{-8}&\color{blue}{3}&\color{blue}{9}&\color{blue}{3}&\color{blue}{8}&\color{orangered}{-1} \end{array} $$The solution is:
$$ \frac{ 6x^{6}+46x^{5}-69x^{4}+36x^{3}+84x^{2}+35x+71 }{ x+9 } = \color{blue}{6x^{5}-8x^{4}+3x^{3}+9x^{2}+3x+8} \color{red}{~-~} \frac{ \color{red}{ 1 } }{ x+9 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 9 = 0 $ ( $ x = \color{blue}{ -9 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-9}&6&46&-69&36&84&35&71\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-9&\color{orangered}{ 6 }&46&-69&36&84&35&71\\& & & & & & & \\ \hline &\color{orangered}{6}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 6 } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-9}&6&46&-69&36&84&35&71\\& & \color{blue}{-54} & & & & & \\ \hline &\color{blue}{6}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 46 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrrrr}-9&6&\color{orangered}{ 46 }&-69&36&84&35&71\\& & \color{orangered}{-54} & & & & & \\ \hline &6&\color{orangered}{-8}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-9}&6&46&-69&36&84&35&71\\& & -54& \color{blue}{72} & & & & \\ \hline &6&\color{blue}{-8}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -69 } + \color{orangered}{ 72 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrrr}-9&6&46&\color{orangered}{ -69 }&36&84&35&71\\& & -54& \color{orangered}{72} & & & & \\ \hline &6&-8&\color{orangered}{3}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 3 } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-9}&6&46&-69&36&84&35&71\\& & -54& 72& \color{blue}{-27} & & & \\ \hline &6&-8&\color{blue}{3}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 36 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrrrr}-9&6&46&-69&\color{orangered}{ 36 }&84&35&71\\& & -54& 72& \color{orangered}{-27} & & & \\ \hline &6&-8&3&\color{orangered}{9}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 9 } = \color{blue}{ -81 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-9}&6&46&-69&36&84&35&71\\& & -54& 72& -27& \color{blue}{-81} & & \\ \hline &6&-8&3&\color{blue}{9}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 84 } + \color{orangered}{ \left( -81 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrrr}-9&6&46&-69&36&\color{orangered}{ 84 }&35&71\\& & -54& 72& -27& \color{orangered}{-81} & & \\ \hline &6&-8&3&9&\color{orangered}{3}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 3 } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-9}&6&46&-69&36&84&35&71\\& & -54& 72& -27& -81& \color{blue}{-27} & \\ \hline &6&-8&3&9&\color{blue}{3}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 35 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrrr}-9&6&46&-69&36&84&\color{orangered}{ 35 }&71\\& & -54& 72& -27& -81& \color{orangered}{-27} & \\ \hline &6&-8&3&9&3&\color{orangered}{8}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 8 } = \color{blue}{ -72 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-9}&6&46&-69&36&84&35&71\\& & -54& 72& -27& -81& -27& \color{blue}{-72} \\ \hline &6&-8&3&9&3&\color{blue}{8}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 71 } + \color{orangered}{ \left( -72 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrrr}-9&6&46&-69&36&84&35&\color{orangered}{ 71 }\\& & -54& 72& -27& -81& -27& \color{orangered}{-72} \\ \hline &\color{blue}{6}&\color{blue}{-8}&\color{blue}{3}&\color{blue}{9}&\color{blue}{3}&\color{blue}{8}&\color{orangered}{-1} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{5}-8x^{4}+3x^{3}+9x^{2}+3x+8 } $ with a remainder of $ \color{red}{ -1 } $.