Divide using synthetic division $$ ( 8x^{3}-8x^{2}+2x-5202 ) \div ( x-9 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}9&8&-8&2&-5202\\& & 72& 576& \color{black}{5202} \\ \hline &\color{blue}{8}&\color{blue}{64}&\color{blue}{578}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ 8x^{3}-8x^{2}+2x-5202 }{ x-9 } = \color{blue}{8x^{2}+64x+578} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{9}&8&-8&2&-5202\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}9&\color{orangered}{ 8 }&-8&2&-5202\\& & & & \\ \hline &\color{orangered}{8}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 8 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&8&-8&2&-5202\\& & \color{blue}{72} & & \\ \hline &\color{blue}{8}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 72 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrrr}9&8&\color{orangered}{ -8 }&2&-5202\\& & \color{orangered}{72} & & \\ \hline &8&\color{orangered}{64}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 64 } = \color{blue}{ 576 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&8&-8&2&-5202\\& & 72& \color{blue}{576} & \\ \hline &8&\color{blue}{64}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 576 } = \color{orangered}{ 578 } $
$$ \begin{array}{c|rrrr}9&8&-8&\color{orangered}{ 2 }&-5202\\& & 72& \color{orangered}{576} & \\ \hline &8&64&\color{orangered}{578}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ 578 } = \color{blue}{ 5202 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&8&-8&2&-5202\\& & 72& 576& \color{blue}{5202} \\ \hline &8&64&\color{blue}{578}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -5202 } + \color{orangered}{ 5202 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}9&8&-8&2&\color{orangered}{ -5202 }\\& & 72& 576& \color{orangered}{5202} \\ \hline &\color{blue}{8}&\color{blue}{64}&\color{blue}{578}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}+64x+578 } $ with a remainder of $ \color{red}{ 0 } $.