Divide using synthetic division $$ ( -10x^{2}-71x-9 ) \div ( -9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}0&-10&-71&-9\\& & 0& \color{black}{0} \\ \hline &\color{blue}{-10}&\color{blue}{-71}&\color{orangered}{-9} \end{array} $$The solution is:
$$ \frac{ -10x^{2}-71x-9 }{ x } = \color{blue}{-10x-71} \color{red}{~-~} \frac{ \color{red}{ 9 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrr}\color{blue}{0}&-10&-71&-9\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}0&\color{orangered}{ -10 }&-71&-9\\& & & \\ \hline &\color{orangered}{-10}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&-10&-71&-9\\& & \color{blue}{0} & \\ \hline &\color{blue}{-10}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -71 } + \color{orangered}{ 0 } = \color{orangered}{ -71 } $
$$ \begin{array}{c|rrr}0&-10&\color{orangered}{ -71 }&-9\\& & \color{orangered}{0} & \\ \hline &-10&\color{orangered}{-71}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -71 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&-10&-71&-9\\& & 0& \color{blue}{0} \\ \hline &-10&\color{blue}{-71}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 0 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrr}0&-10&-71&\color{orangered}{ -9 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{-10}&\color{blue}{-71}&\color{orangered}{-9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -10x-71 } $ with a remainder of $ \color{red}{ -9 } $.