Divide using synthetic division $$ ( 5x^{2}+3x-12 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&5&3&-12\\& & 15& \color{black}{54} \\ \hline &\color{blue}{5}&\color{blue}{18}&\color{orangered}{42} \end{array} $$The solution is:
$$ \frac{ 5x^{2}+3x-12 }{ x-3 } = \color{blue}{5x+18} ~+~ \frac{ \color{red}{ 42 } }{ x-3 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{3}&5&3&-12\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ 5 }&3&-12\\& & & \\ \hline &\color{orangered}{5}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 5 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&5&3&-12\\& & \color{blue}{15} & \\ \hline &\color{blue}{5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 15 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrr}3&5&\color{orangered}{ 3 }&-12\\& & \color{orangered}{15} & \\ \hline &5&\color{orangered}{18}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 18 } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&5&3&-12\\& & 15& \color{blue}{54} \\ \hline &5&\color{blue}{18}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 54 } = \color{orangered}{ 42 } $
$$ \begin{array}{c|rrr}3&5&3&\color{orangered}{ -12 }\\& & 15& \color{orangered}{54} \\ \hline &\color{blue}{5}&\color{blue}{18}&\color{orangered}{42} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x+18 } $ with a remainder of $ \color{red}{ 42 } $.