Divide using synthetic division $$ ( x^{2}-3x-4 ) \div ( -4x-8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}8&1&-3&-4\\& & 8& \color{black}{40} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{orangered}{36} \end{array} $$The solution is:
$$ \frac{ x^{2}-3x-4 }{ x-8 } = \color{blue}{x+5} ~+~ \frac{ \color{red}{ 36 } }{ x-8 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -8 = 0 $ ( $ x = \color{blue}{ 8 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{8}&1&-3&-4\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}8&\color{orangered}{ 1 }&-3&-4\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 1 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrr}\color{blue}{8}&1&-3&-4\\& & \color{blue}{8} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 8 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrr}8&1&\color{orangered}{ -3 }&-4\\& & \color{orangered}{8} & \\ \hline &1&\color{orangered}{5}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 5 } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrr}\color{blue}{8}&1&-3&-4\\& & 8& \color{blue}{40} \\ \hline &1&\color{blue}{5}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 40 } = \color{orangered}{ 36 } $
$$ \begin{array}{c|rrr}8&1&-3&\color{orangered}{ -4 }\\& & 8& \color{orangered}{40} \\ \hline &\color{blue}{1}&\color{blue}{5}&\color{orangered}{36} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x+5 } $ with a remainder of $ \color{red}{ 36 } $.