Divide using synthetic division $$ ( 9x^{4}-9x^{3}-58x^{2}+5x+24 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-1&9&-9&-58&5&24\\& & -9& 18& 40& \color{black}{-45} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{blue}{-40}&\color{blue}{45}&\color{orangered}{-21} \end{array} $$The solution is:
$$ \frac{ 9x^{4}-9x^{3}-58x^{2}+5x+24 }{ x+1 } = \color{blue}{9x^{3}-18x^{2}-40x+45} \color{red}{~-~} \frac{ \color{red}{ 21 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&9&-9&-58&5&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ 9 }&-9&-58&5&24\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 9 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&9&-9&-58&5&24\\& & \color{blue}{-9} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrr}-1&9&\color{orangered}{ -9 }&-58&5&24\\& & \color{orangered}{-9} & & & \\ \hline &9&\color{orangered}{-18}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&9&-9&-58&5&24\\& & -9& \color{blue}{18} & & \\ \hline &9&\color{blue}{-18}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -58 } + \color{orangered}{ 18 } = \color{orangered}{ -40 } $
$$ \begin{array}{c|rrrrr}-1&9&-9&\color{orangered}{ -58 }&5&24\\& & -9& \color{orangered}{18} & & \\ \hline &9&-18&\color{orangered}{-40}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -40 \right) } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&9&-9&-58&5&24\\& & -9& 18& \color{blue}{40} & \\ \hline &9&-18&\color{blue}{-40}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 40 } = \color{orangered}{ 45 } $
$$ \begin{array}{c|rrrrr}-1&9&-9&-58&\color{orangered}{ 5 }&24\\& & -9& 18& \color{orangered}{40} & \\ \hline &9&-18&-40&\color{orangered}{45}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 45 } = \color{blue}{ -45 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&9&-9&-58&5&24\\& & -9& 18& 40& \color{blue}{-45} \\ \hline &9&-18&-40&\color{blue}{45}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -45 \right) } = \color{orangered}{ -21 } $
$$ \begin{array}{c|rrrrr}-1&9&-9&-58&5&\color{orangered}{ 24 }\\& & -9& 18& 40& \color{orangered}{-45} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{blue}{-40}&\color{blue}{45}&\color{orangered}{-21} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}-18x^{2}-40x+45 } $ with a remainder of $ \color{red}{ -21 } $.