Check if $ x-3 $ is a factor of $ 11x^{2}-12x-4 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&11&-12&-4\\& & 33& \color{black}{63} \\ \hline &\color{blue}{11}&\color{blue}{21}&\color{orangered}{59} \end{array} $$Because the remainder $ \left( \color{red}{ 59 } \right) $ is not zero, we conclude that the $ x-3 $ is not a factor of $ 11x^{2}-12x-4$.
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 -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{3}&11&-12&-4\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ 11 }&-12&-4\\& & & \\ \hline &\color{orangered}{11}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 11 } = \color{blue}{ 33 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&11&-12&-4\\& & \color{blue}{33} & \\ \hline &\color{blue}{11}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 33 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrr}3&11&\color{orangered}{ -12 }&-4\\& & \color{orangered}{33} & \\ \hline &11&\color{orangered}{21}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 21 } = \color{blue}{ 63 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&11&-12&-4\\& & 33& \color{blue}{63} \\ \hline &11&\color{blue}{21}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 63 } = \color{orangered}{ 59 } $
$$ \begin{array}{c|rrr}3&11&-12&\color{orangered}{ -4 }\\& & 33& \color{orangered}{63} \\ \hline &\color{blue}{11}&\color{blue}{21}&\color{orangered}{59} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 59 }\right)$.