Divide using synthetic division $$ ( -7x^{2}-7x+20 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&-7&-7&20\\& & 28& \color{black}{-84} \\ \hline &\color{blue}{-7}&\color{blue}{21}&\color{orangered}{-64} \end{array} $$The solution is:
$$ \frac{ -7x^{2}-7x+20 }{ x+4 } = \color{blue}{-7x+21} \color{red}{~-~} \frac{ \color{red}{ 64 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-7&-7&20\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ -7 }&-7&20\\& & & \\ \hline &\color{orangered}{-7}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-7&-7&20\\& & \color{blue}{28} & \\ \hline &\color{blue}{-7}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 28 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrr}-4&-7&\color{orangered}{ -7 }&20\\& & \color{orangered}{28} & \\ \hline &-7&\color{orangered}{21}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 21 } = \color{blue}{ -84 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-7&-7&20\\& & 28& \color{blue}{-84} \\ \hline &-7&\color{blue}{21}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -84 \right) } = \color{orangered}{ -64 } $
$$ \begin{array}{c|rrr}-4&-7&-7&\color{orangered}{ 20 }\\& & 28& \color{orangered}{-84} \\ \hline &\color{blue}{-7}&\color{blue}{21}&\color{orangered}{-64} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -7x+21 } $ with a remainder of $ \color{red}{ -64 } $.