Check if $ 3x+4 $ is a factor of $ 9x^{4}-6x^{3}-2x^{2}+16x+8 $.
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&9&-6&-2&16&8\\& & -36& 168& -664& \color{black}{2592} \\ \hline &\color{blue}{9}&\color{blue}{-42}&\color{blue}{166}&\color{blue}{-648}&\color{orangered}{2600} \end{array} $$Because the remainder $ \left( \color{red}{ 2600 } \right) $ is not zero, we conclude that the $ x+4 $ is not a factor of $ 9x^{4}-6x^{3}-2x^{2}+16x+8$.
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 + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&9&-6&-2&16&8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 9 }&-6&-2&16&8\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 9 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&9&-6&-2&16&8\\& & \color{blue}{-36} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -42 } $
$$ \begin{array}{c|rrrrr}-4&9&\color{orangered}{ -6 }&-2&16&8\\& & \color{orangered}{-36} & & & \\ \hline &9&\color{orangered}{-42}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -42 \right) } = \color{blue}{ 168 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&9&-6&-2&16&8\\& & -36& \color{blue}{168} & & \\ \hline &9&\color{blue}{-42}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 168 } = \color{orangered}{ 166 } $
$$ \begin{array}{c|rrrrr}-4&9&-6&\color{orangered}{ -2 }&16&8\\& & -36& \color{orangered}{168} & & \\ \hline &9&-42&\color{orangered}{166}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 166 } = \color{blue}{ -664 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&9&-6&-2&16&8\\& & -36& 168& \color{blue}{-664} & \\ \hline &9&-42&\color{blue}{166}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -664 \right) } = \color{orangered}{ -648 } $
$$ \begin{array}{c|rrrrr}-4&9&-6&-2&\color{orangered}{ 16 }&8\\& & -36& 168& \color{orangered}{-664} & \\ \hline &9&-42&166&\color{orangered}{-648}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -648 \right) } = \color{blue}{ 2592 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&9&-6&-2&16&8\\& & -36& 168& -664& \color{blue}{2592} \\ \hline &9&-42&166&\color{blue}{-648}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 2592 } = \color{orangered}{ 2600 } $
$$ \begin{array}{c|rrrrr}-4&9&-6&-2&16&\color{orangered}{ 8 }\\& & -36& 168& -664& \color{orangered}{2592} \\ \hline &\color{blue}{9}&\color{blue}{-42}&\color{blue}{166}&\color{blue}{-648}&\color{orangered}{2600} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 2600 }\right)$.