Divide using synthetic division $$ ( 12x^{2}-13x-9 ) \div ( 4x-7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}7&12&-13&-9\\& & 84& \color{black}{497} \\ \hline &\color{blue}{12}&\color{blue}{71}&\color{orangered}{488} \end{array} $$The solution is:
$$ \frac{ 12x^{2}-13x-9 }{ x-7 } = \color{blue}{12x+71} ~+~ \frac{ \color{red}{ 488 } }{ x-7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{7}&12&-13&-9\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}7&\color{orangered}{ 12 }&-13&-9\\& & & \\ \hline &\color{orangered}{12}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 12 } = \color{blue}{ 84 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&12&-13&-9\\& & \color{blue}{84} & \\ \hline &\color{blue}{12}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 84 } = \color{orangered}{ 71 } $
$$ \begin{array}{c|rrr}7&12&\color{orangered}{ -13 }&-9\\& & \color{orangered}{84} & \\ \hline &12&\color{orangered}{71}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 71 } = \color{blue}{ 497 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&12&-13&-9\\& & 84& \color{blue}{497} \\ \hline &12&\color{blue}{71}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 497 } = \color{orangered}{ 488 } $
$$ \begin{array}{c|rrr}7&12&-13&\color{orangered}{ -9 }\\& & 84& \color{orangered}{497} \\ \hline &\color{blue}{12}&\color{blue}{71}&\color{orangered}{488} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 12x+71 } $ with a remainder of $ \color{red}{ 488 } $.