Divide using synthetic division $$ ( x^{2}-2x-3 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-6&1&-2&-3\\& & -6& \color{black}{48} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{orangered}{45} \end{array} $$The solution is:
$$ \frac{ x^{2}-2x-3 }{ x+6 } = \color{blue}{x-8} ~+~ \frac{ \color{red}{ 45 } }{ x+6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-6}&1&-2&-3\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-6&\color{orangered}{ 1 }&-2&-3\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 1 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&1&-2&-3\\& & \color{blue}{-6} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrr}-6&1&\color{orangered}{ -2 }&-3\\& & \color{orangered}{-6} & \\ \hline &1&\color{orangered}{-8}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&1&-2&-3\\& & -6& \color{blue}{48} \\ \hline &1&\color{blue}{-8}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 48 } = \color{orangered}{ 45 } $
$$ \begin{array}{c|rrr}-6&1&-2&\color{orangered}{ -3 }\\& & -6& \color{orangered}{48} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{orangered}{45} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x-8 } $ with a remainder of $ \color{red}{ 45 } $.