Divide using synthetic division $$ ( -9x^{2}-72x ) \div ( x+8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-8&-9&-72&0\\& & 72& \color{black}{0} \\ \hline &\color{blue}{-9}&\color{blue}{0}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -9x^{2}-72x }{ x+8 } = \color{blue}{-9x} $$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.
$$ \begin{array}{c|rrr}\color{blue}{-8}&-9&-72&0\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-8&\color{orangered}{ -9 }&-72&0\\& & & \\ \hline &\color{orangered}{-9}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrr}\color{blue}{-8}&-9&-72&0\\& & \color{blue}{72} & \\ \hline &\color{blue}{-9}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -72 } + \color{orangered}{ 72 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}-8&-9&\color{orangered}{ -72 }&0\\& & \color{orangered}{72} & \\ \hline &-9&\color{orangered}{0}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{-8}&-9&-72&0\\& & 72& \color{blue}{0} \\ \hline &-9&\color{blue}{0}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}-8&-9&-72&\color{orangered}{ 0 }\\& & 72& \color{orangered}{0} \\ \hline &\color{blue}{-9}&\color{blue}{0}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -9x } $ with a remainder of $ \color{red}{ 0 } $.