Divide using synthetic division $$ ( 11x^{2}-12x-4 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&11&-12&-4\\& & 33& \color{black}{63} \\ \hline &\color{blue}{11}&\color{blue}{21}&\color{orangered}{59} \end{array} $$The solution is:
$$ \frac{ 11x^{2}-12x-4 }{ x-3 } = \color{blue}{11x+21} ~+~ \frac{ \color{red}{ 59 } }{ x-3 } $$Explanation
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 } $
$$ \begin{array}{c|rrr}3&11&-12&\color{orangered}{ -4 }\\& & 33& \color{orangered}{63} \\ \hline &\color{blue}{11}&\color{blue}{21}&\color{orangered}{59} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 11x+21 } $ with a remainder of $ \color{red}{ 59 } $.