Divide using synthetic division $$ ( 15x^{2}+4x-4 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&15&4&-4\\& & 45& \color{black}{147} \\ \hline &\color{blue}{15}&\color{blue}{49}&\color{orangered}{143} \end{array} $$The solution is:
$$ \frac{ 15x^{2}+4x-4 }{ x-3 } = \color{blue}{15x+49} ~+~ \frac{ \color{red}{ 143 } }{ x-3 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{3}&15&4&-4\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ 15 }&4&-4\\& & & \\ \hline &\color{orangered}{15}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 15 } = \color{blue}{ 45 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&15&4&-4\\& & \color{blue}{45} & \\ \hline &\color{blue}{15}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 45 } = \color{orangered}{ 49 } $
$$ \begin{array}{c|rrr}3&15&\color{orangered}{ 4 }&-4\\& & \color{orangered}{45} & \\ \hline &15&\color{orangered}{49}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 49 } = \color{blue}{ 147 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&15&4&-4\\& & 45& \color{blue}{147} \\ \hline &15&\color{blue}{49}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 147 } = \color{orangered}{ 143 } $
$$ \begin{array}{c|rrr}3&15&4&\color{orangered}{ -4 }\\& & 45& \color{orangered}{147} \\ \hline &\color{blue}{15}&\color{blue}{49}&\color{orangered}{143} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 15x+49 } $ with a remainder of $ \color{red}{ 143 } $.