Divide using synthetic division $$ ( x^{2}-x-42 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-6&1&-1&-42\\& & -6& \color{black}{42} \\ \hline &\color{blue}{1}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{2}-x-42 }{ x+6 } = \color{blue}{x-7} $$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.
$$ \begin{array}{c|rrr}\color{blue}{-6}&1&-1&-42\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-6&\color{orangered}{ 1 }&-1&-42\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&1&-1&-42\\& & \color{blue}{-6} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrr}-6&1&\color{orangered}{ -1 }&-42\\& & \color{orangered}{-6} & \\ \hline &1&\color{orangered}{-7}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&1&-1&-42\\& & -6& \color{blue}{42} \\ \hline &1&\color{blue}{-7}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -42 } + \color{orangered}{ 42 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}-6&1&-1&\color{orangered}{ -42 }\\& & -6& \color{orangered}{42} \\ \hline &\color{blue}{1}&\color{blue}{-7}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x-7 } $ with a remainder of $ \color{red}{ 0 } $.