Divide using synthetic division $$ ( x^{2}+12x-15 ) \div ( x-7 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}7&1&12&-15\\& & 7& \color{black}{133} \\ \hline &\color{blue}{1}&\color{blue}{19}&\color{orangered}{118} \end{array} $$The solution is:
$$ \dfrac{ x^{2}+12x-15 }{ x-7 } = \color{blue}{x+19} ~+~ \dfrac{ \color{red}{ 118 } }{ x-7 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{7}&1&12&-15\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}7&\color{orangered}{ 1 }&12&-15\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 1 } = \color{blue}{ 7 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&1&12&-15\\& & \color{blue}{7} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ 7 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrr}7&1&\color{orangered}{ 12 }&-15\\& & \color{orangered}{7} & \\ \hline &1&\color{orangered}{19}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 19 } = \color{blue}{ 133 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&1&12&-15\\& & 7& \color{blue}{133} \\ \hline &1&\color{blue}{19}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 133 } = \color{orangered}{ 118 } $
$$ \begin{array}{c|rrr}7&1&12&\color{orangered}{ -15 }\\& & 7& \color{orangered}{133} \\ \hline &\color{blue}{1}&\color{blue}{19}&\color{orangered}{118} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x+19 } $ with a remainder of $ \color{red}{ 118 } $.