Check if $ x+9 $ is a factor of $ x^{2}+2x-63 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-9&1&2&-63\\& & -9& \color{black}{63} \\ \hline &\color{blue}{1}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$Because the remainder equals zero, we conclude that the $ x+9 $ is a factor of the $ x^{2}+2x-63 $.
Explanation
First we need to create a synthetic division table.
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 9 = 0 $ ( $ x = \color{blue}{ -9 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-9}&1&2&-63\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-9&\color{orangered}{ 1 }&2&-63\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 1 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrr}\color{blue}{-9}&1&2&-63\\& & \color{blue}{-9} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrr}-9&1&\color{orangered}{ 2 }&-63\\& & \color{orangered}{-9} & \\ \hline &1&\color{orangered}{-7}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 63 } $.
$$ \begin{array}{c|rrr}\color{blue}{-9}&1&2&-63\\& & -9& \color{blue}{63} \\ \hline &1&\color{blue}{-7}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -63 } + \color{orangered}{ 63 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}-9&1&2&\color{orangered}{ -63 }\\& & -9& \color{orangered}{63} \\ \hline &\color{blue}{1}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right)$.