Divide using synthetic division $$ ( 9x^{4}-9x^{3}-58x^{2}+5x+24 ) \div ( x-6 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}6&9&-9&-58&5&24\\& & 54& 270& 1272& \color{black}{7662} \\ \hline &\color{blue}{9}&\color{blue}{45}&\color{blue}{212}&\color{blue}{1277}&\color{orangered}{7686} \end{array} $$The solution is:
$$ \dfrac{ 9x^{4}-9x^{3}-58x^{2}+5x+24 }{ x-6 } = \color{blue}{9x^{3}+45x^{2}+212x+1277} ~+~ \dfrac{ \color{red}{ 7686 } }{ x-6 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&9&-9&-58&5&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}6&\color{orangered}{ 9 }&-9&-58&5&24\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 9 } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&9&-9&-58&5&24\\& & \color{blue}{54} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 54 } = \color{orangered}{ 45 } $
$$ \begin{array}{c|rrrrr}6&9&\color{orangered}{ -9 }&-58&5&24\\& & \color{orangered}{54} & & & \\ \hline &9&\color{orangered}{45}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 45 } = \color{blue}{ 270 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&9&-9&-58&5&24\\& & 54& \color{blue}{270} & & \\ \hline &9&\color{blue}{45}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -58 } + \color{orangered}{ 270 } = \color{orangered}{ 212 } $
$$ \begin{array}{c|rrrrr}6&9&-9&\color{orangered}{ -58 }&5&24\\& & 54& \color{orangered}{270} & & \\ \hline &9&45&\color{orangered}{212}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 212 } = \color{blue}{ 1272 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&9&-9&-58&5&24\\& & 54& 270& \color{blue}{1272} & \\ \hline &9&45&\color{blue}{212}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 1272 } = \color{orangered}{ 1277 } $
$$ \begin{array}{c|rrrrr}6&9&-9&-58&\color{orangered}{ 5 }&24\\& & 54& 270& \color{orangered}{1272} & \\ \hline &9&45&212&\color{orangered}{1277}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 1277 } = \color{blue}{ 7662 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&9&-9&-58&5&24\\& & 54& 270& 1272& \color{blue}{7662} \\ \hline &9&45&212&\color{blue}{1277}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 7662 } = \color{orangered}{ 7686 } $
$$ \begin{array}{c|rrrrr}6&9&-9&-58&5&\color{orangered}{ 24 }\\& & 54& 270& 1272& \color{orangered}{7662} \\ \hline &\color{blue}{9}&\color{blue}{45}&\color{blue}{212}&\color{blue}{1277}&\color{orangered}{7686} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}+45x^{2}+212x+1277 } $ with a remainder of $ \color{red}{ 7686 } $.