Divide using synthetic division $$ ( 9x^{4}-9x^{3}-58x^{2}+5x+24 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&9&-9&-58&5&24\\& & -36& 180& -488& \color{black}{1932} \\ \hline &\color{blue}{9}&\color{blue}{-45}&\color{blue}{122}&\color{blue}{-483}&\color{orangered}{1956} \end{array} $$The solution is:
$$ \frac{ 9x^{4}-9x^{3}-58x^{2}+5x+24 }{ x+4 } = \color{blue}{9x^{3}-45x^{2}+122x-483} ~+~ \frac{ \color{red}{ 1956 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&9&-9&-58&5&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 9 }&-9&-58&5&24\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 9 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&9&-9&-58&5&24\\& & \color{blue}{-36} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -45 } $
$$ \begin{array}{c|rrrrr}-4&9&\color{orangered}{ -9 }&-58&5&24\\& & \color{orangered}{-36} & & & \\ \hline &9&\color{orangered}{-45}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -45 \right) } = \color{blue}{ 180 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&9&-9&-58&5&24\\& & -36& \color{blue}{180} & & \\ \hline &9&\color{blue}{-45}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -58 } + \color{orangered}{ 180 } = \color{orangered}{ 122 } $
$$ \begin{array}{c|rrrrr}-4&9&-9&\color{orangered}{ -58 }&5&24\\& & -36& \color{orangered}{180} & & \\ \hline &9&-45&\color{orangered}{122}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 122 } = \color{blue}{ -488 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&9&-9&-58&5&24\\& & -36& 180& \color{blue}{-488} & \\ \hline &9&-45&\color{blue}{122}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -488 \right) } = \color{orangered}{ -483 } $
$$ \begin{array}{c|rrrrr}-4&9&-9&-58&\color{orangered}{ 5 }&24\\& & -36& 180& \color{orangered}{-488} & \\ \hline &9&-45&122&\color{orangered}{-483}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -483 \right) } = \color{blue}{ 1932 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&9&-9&-58&5&24\\& & -36& 180& -488& \color{blue}{1932} \\ \hline &9&-45&122&\color{blue}{-483}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 1932 } = \color{orangered}{ 1956 } $
$$ \begin{array}{c|rrrrr}-4&9&-9&-58&5&\color{orangered}{ 24 }\\& & -36& 180& -488& \color{orangered}{1932} \\ \hline &\color{blue}{9}&\color{blue}{-45}&\color{blue}{122}&\color{blue}{-483}&\color{orangered}{1956} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}-45x^{2}+122x-483 } $ with a remainder of $ \color{red}{ 1956 } $.