Divide using synthetic division $$ ( 8x^{3}-8x^{2}+2x-5202 ) \div ( x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-9&8&-8&2&-5202\\& & -72& 720& \color{black}{-6498} \\ \hline &\color{blue}{8}&\color{blue}{-80}&\color{blue}{722}&\color{orangered}{-11700} \end{array} $$The solution is:
$$ \frac{ 8x^{3}-8x^{2}+2x-5202 }{ x+9 } = \color{blue}{8x^{2}-80x+722} \color{red}{~-~} \frac{ \color{red}{ 11700 } }{ x+9 } $$Explanation
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}{ \left( -72 \right) } = \color{orangered}{ -80 } $
$$ \begin{array}{c|rrrr}-9&8&\color{orangered}{ -8 }&2&-5202\\& & \color{orangered}{-72} & & \\ \hline &8&\color{orangered}{-80}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -80 \right) } = \color{blue}{ 720 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&8&-8&2&-5202\\& & -72& \color{blue}{720} & \\ \hline &8&\color{blue}{-80}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 720 } = \color{orangered}{ 722 } $
$$ \begin{array}{c|rrrr}-9&8&-8&\color{orangered}{ 2 }&-5202\\& & -72& \color{orangered}{720} & \\ \hline &8&-80&\color{orangered}{722}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 722 } = \color{blue}{ -6498 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&8&-8&2&-5202\\& & -72& 720& \color{blue}{-6498} \\ \hline &8&-80&\color{blue}{722}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -5202 } + \color{orangered}{ \left( -6498 \right) } = \color{orangered}{ -11700 } $
$$ \begin{array}{c|rrrr}-9&8&-8&2&\color{orangered}{ -5202 }\\& & -72& 720& \color{orangered}{-6498} \\ \hline &\color{blue}{8}&\color{blue}{-80}&\color{blue}{722}&\color{orangered}{-11700} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{2}-80x+722 } $ with a remainder of $ \color{red}{ -11700 } $.