Divide using synthetic division $$ ( 9x^{4}-9x^{3}-58x^{2}+5x+24 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&9&-9&-58&5&24\\& & -18& 54& 8& \color{black}{-26} \\ \hline &\color{blue}{9}&\color{blue}{-27}&\color{blue}{-4}&\color{blue}{13}&\color{orangered}{-2} \end{array} $$The solution is:
$$ \frac{ 9x^{4}-9x^{3}-58x^{2}+5x+24 }{ x+2 } = \color{blue}{9x^{3}-27x^{2}-4x+13} \color{red}{~-~} \frac{ \color{red}{ 2 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&-9&-58&5&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 9 }&-9&-58&5&24\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 9 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&-9&-58&5&24\\& & \color{blue}{-18} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -27 } $
$$ \begin{array}{c|rrrrr}-2&9&\color{orangered}{ -9 }&-58&5&24\\& & \color{orangered}{-18} & & & \\ \hline &9&\color{orangered}{-27}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -27 \right) } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&-9&-58&5&24\\& & -18& \color{blue}{54} & & \\ \hline &9&\color{blue}{-27}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -58 } + \color{orangered}{ 54 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}-2&9&-9&\color{orangered}{ -58 }&5&24\\& & -18& \color{orangered}{54} & & \\ \hline &9&-27&\color{orangered}{-4}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&-9&-58&5&24\\& & -18& 54& \color{blue}{8} & \\ \hline &9&-27&\color{blue}{-4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 8 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrrr}-2&9&-9&-58&\color{orangered}{ 5 }&24\\& & -18& 54& \color{orangered}{8} & \\ \hline &9&-27&-4&\color{orangered}{13}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 13 } = \color{blue}{ -26 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&-9&-58&5&24\\& & -18& 54& 8& \color{blue}{-26} \\ \hline &9&-27&-4&\color{blue}{13}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -26 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-2&9&-9&-58&5&\color{orangered}{ 24 }\\& & -18& 54& 8& \color{orangered}{-26} \\ \hline &\color{blue}{9}&\color{blue}{-27}&\color{blue}{-4}&\color{blue}{13}&\color{orangered}{-2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}-27x^{2}-4x+13 } $ with a remainder of $ \color{red}{ -2 } $.