Divide using synthetic division $$ ( -20x^{2}-21x-6 ) \div ( 5x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&-20&-21&-6\\& & 80& \color{black}{-236} \\ \hline &\color{blue}{-20}&\color{blue}{59}&\color{orangered}{-242} \end{array} $$The solution is:
$$ \frac{ -20x^{2}-21x-6 }{ x+4 } = \color{blue}{-20x+59} \color{red}{~-~} \frac{ \color{red}{ 242 } }{ x+4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-20&-21&-6\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ -20 }&-21&-6\\& & & \\ \hline &\color{orangered}{-20}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -20 \right) } = \color{blue}{ 80 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-20&-21&-6\\& & \color{blue}{80} & \\ \hline &\color{blue}{-20}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 80 } = \color{orangered}{ 59 } $
$$ \begin{array}{c|rrr}-4&-20&\color{orangered}{ -21 }&-6\\& & \color{orangered}{80} & \\ \hline &-20&\color{orangered}{59}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 59 } = \color{blue}{ -236 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-20&-21&-6\\& & 80& \color{blue}{-236} \\ \hline &-20&\color{blue}{59}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -236 \right) } = \color{orangered}{ -242 } $
$$ \begin{array}{c|rrr}-4&-20&-21&\color{orangered}{ -6 }\\& & 80& \color{orangered}{-236} \\ \hline &\color{blue}{-20}&\color{blue}{59}&\color{orangered}{-242} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -20x+59 } $ with a remainder of $ \color{red}{ -242 } $.