Divide using synthetic division $$ ( x^{2}+2x-99 ) \div ( x+11 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-11&1&2&-99\\& & -11& \color{black}{99} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{2}+2x-99 }{ x+11 } = \color{blue}{x-9} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 11 = 0 $ ( $ x = \color{blue}{ -11 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-11}&1&2&-99\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-11&\color{orangered}{ 1 }&2&-99\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -11 } \cdot \color{blue}{ 1 } = \color{blue}{ -11 } $.
$$ \begin{array}{c|rrr}\color{blue}{-11}&1&2&-99\\& & \color{blue}{-11} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -11 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrr}-11&1&\color{orangered}{ 2 }&-99\\& & \color{orangered}{-11} & \\ \hline &1&\color{orangered}{-9}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -11 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 99 } $.
$$ \begin{array}{c|rrr}\color{blue}{-11}&1&2&-99\\& & -11& \color{blue}{99} \\ \hline &1&\color{blue}{-9}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -99 } + \color{orangered}{ 99 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}-11&1&2&\color{orangered}{ -99 }\\& & -11& \color{orangered}{99} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x-9 } $ with a remainder of $ \color{red}{ 0 } $.