Divide using synthetic division $$ ( -x^{2}-65x+20 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}9&-1&-65&20\\& & -9& \color{black}{-666} \\ \hline &\color{blue}{-1}&\color{blue}{-74}&\color{orangered}{-646} \end{array} $$The solution is:
$$ \frac{ -x^{2}-65x+20 }{ x-9 } = \color{blue}{-x-74} \color{red}{~-~} \frac{ \color{red}{ 646 } }{ x-9 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -9 = 0 $ ( $ x = \color{blue}{ 9 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{9}&-1&-65&20\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}9&\color{orangered}{ -1 }&-65&20\\& & & \\ \hline &\color{orangered}{-1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrr}\color{blue}{9}&-1&-65&20\\& & \color{blue}{-9} & \\ \hline &\color{blue}{-1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -65 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -74 } $
$$ \begin{array}{c|rrr}9&-1&\color{orangered}{ -65 }&20\\& & \color{orangered}{-9} & \\ \hline &-1&\color{orangered}{-74}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -74 \right) } = \color{blue}{ -666 } $.
$$ \begin{array}{c|rrr}\color{blue}{9}&-1&-65&20\\& & -9& \color{blue}{-666} \\ \hline &-1&\color{blue}{-74}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -666 \right) } = \color{orangered}{ -646 } $
$$ \begin{array}{c|rrr}9&-1&-65&\color{orangered}{ 20 }\\& & -9& \color{orangered}{-666} \\ \hline &\color{blue}{-1}&\color{blue}{-74}&\color{orangered}{-646} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x-74 } $ with a remainder of $ \color{red}{ -646 } $.