Check if $ x+8 $ is a factor of $ 9x^{4}-6x^{3}-2x^{2}+16x+8 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-8&9&-6&-2&16&8\\& & -72& 624& -4976& \color{black}{39680} \\ \hline &\color{blue}{9}&\color{blue}{-78}&\color{blue}{622}&\color{blue}{-4960}&\color{orangered}{39688} \end{array} $$Because the remainder $ \left( \color{red}{ 39688 } \right) $ is not zero, we conclude that the $ x+8 $ is not a factor of $ 9x^{4}-6x^{3}-2x^{2}+16x+8$.
Explanation
First we need to create a synthetic division table.
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 8 = 0 $ ( $ x = \color{blue}{ -8 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-8}&9&-6&-2&16&8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-8&\color{orangered}{ 9 }&-6&-2&16&8\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 9 } = \color{blue}{ -72 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-8}&9&-6&-2&16&8\\& & \color{blue}{-72} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -72 \right) } = \color{orangered}{ -78 } $
$$ \begin{array}{c|rrrrr}-8&9&\color{orangered}{ -6 }&-2&16&8\\& & \color{orangered}{-72} & & & \\ \hline &9&\color{orangered}{-78}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -78 \right) } = \color{blue}{ 624 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-8}&9&-6&-2&16&8\\& & -72& \color{blue}{624} & & \\ \hline &9&\color{blue}{-78}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 624 } = \color{orangered}{ 622 } $
$$ \begin{array}{c|rrrrr}-8&9&-6&\color{orangered}{ -2 }&16&8\\& & -72& \color{orangered}{624} & & \\ \hline &9&-78&\color{orangered}{622}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 622 } = \color{blue}{ -4976 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-8}&9&-6&-2&16&8\\& & -72& 624& \color{blue}{-4976} & \\ \hline &9&-78&\color{blue}{622}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -4976 \right) } = \color{orangered}{ -4960 } $
$$ \begin{array}{c|rrrrr}-8&9&-6&-2&\color{orangered}{ 16 }&8\\& & -72& 624& \color{orangered}{-4976} & \\ \hline &9&-78&622&\color{orangered}{-4960}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -4960 \right) } = \color{blue}{ 39680 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-8}&9&-6&-2&16&8\\& & -72& 624& -4976& \color{blue}{39680} \\ \hline &9&-78&622&\color{blue}{-4960}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 39680 } = \color{orangered}{ 39688 } $
$$ \begin{array}{c|rrrrr}-8&9&-6&-2&16&\color{orangered}{ 8 }\\& & -72& 624& -4976& \color{orangered}{39680} \\ \hline &\color{blue}{9}&\color{blue}{-78}&\color{blue}{622}&\color{blue}{-4960}&\color{orangered}{39688} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 39688 }\right)$.