Check if $ x-8 $ is a factor of $ 3x^{4}-11x^{3}-3x^{2}-6x+8 $.
The synthetic division table is:
$$ \begin{array}{c|rrrrr}8&3&-11&-3&-6&8\\& & 24& 104& 808& \color{black}{6416} \\ \hline &\color{blue}{3}&\color{blue}{13}&\color{blue}{101}&\color{blue}{802}&\color{orangered}{6424} \end{array} $$Because the remainder $ \left( \color{red}{ 6424 } \right) $ is not zero, we conclude that the $ x-8 $ is not a factor of $ 3x^{4}-11x^{3}-3x^{2}-6x+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 -8 = 0 $ ( $ x = \color{blue}{ 8 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&3&-11&-3&-6&8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}8&\color{orangered}{ 3 }&-11&-3&-6&8\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 3 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&3&-11&-3&-6&8\\& & \color{blue}{24} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 24 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrrr}8&3&\color{orangered}{ -11 }&-3&-6&8\\& & \color{orangered}{24} & & & \\ \hline &3&\color{orangered}{13}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 13 } = \color{blue}{ 104 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&3&-11&-3&-6&8\\& & 24& \color{blue}{104} & & \\ \hline &3&\color{blue}{13}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 104 } = \color{orangered}{ 101 } $
$$ \begin{array}{c|rrrrr}8&3&-11&\color{orangered}{ -3 }&-6&8\\& & 24& \color{orangered}{104} & & \\ \hline &3&13&\color{orangered}{101}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 101 } = \color{blue}{ 808 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&3&-11&-3&-6&8\\& & 24& 104& \color{blue}{808} & \\ \hline &3&13&\color{blue}{101}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 808 } = \color{orangered}{ 802 } $
$$ \begin{array}{c|rrrrr}8&3&-11&-3&\color{orangered}{ -6 }&8\\& & 24& 104& \color{orangered}{808} & \\ \hline &3&13&101&\color{orangered}{802}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 802 } = \color{blue}{ 6416 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&3&-11&-3&-6&8\\& & 24& 104& 808& \color{blue}{6416} \\ \hline &3&13&101&\color{blue}{802}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 6416 } = \color{orangered}{ 6424 } $
$$ \begin{array}{c|rrrrr}8&3&-11&-3&-6&\color{orangered}{ 8 }\\& & 24& 104& 808& \color{orangered}{6416} \\ \hline &\color{blue}{3}&\color{blue}{13}&\color{blue}{101}&\color{blue}{802}&\color{orangered}{6424} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 6424 }\right)$.