Divide using synthetic division $$ ( -11x^{2}+9x-20 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&-11&9&-20\\& & -55& \color{black}{-230} \\ \hline &\color{blue}{-11}&\color{blue}{-46}&\color{orangered}{-250} \end{array} $$The solution is:
$$ \frac{ -11x^{2}+9x-20 }{ x-5 } = \color{blue}{-11x-46} \color{red}{~-~} \frac{ \color{red}{ 250 } }{ x-5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{5}&-11&9&-20\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ -11 }&9&-20\\& & & \\ \hline &\color{orangered}{-11}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ -55 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-11&9&-20\\& & \color{blue}{-55} & \\ \hline &\color{blue}{-11}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -55 \right) } = \color{orangered}{ -46 } $
$$ \begin{array}{c|rrr}5&-11&\color{orangered}{ 9 }&-20\\& & \color{orangered}{-55} & \\ \hline &-11&\color{orangered}{-46}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -46 \right) } = \color{blue}{ -230 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-11&9&-20\\& & -55& \color{blue}{-230} \\ \hline &-11&\color{blue}{-46}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ \left( -230 \right) } = \color{orangered}{ -250 } $
$$ \begin{array}{c|rrr}5&-11&9&\color{orangered}{ -20 }\\& & -55& \color{orangered}{-230} \\ \hline &\color{blue}{-11}&\color{blue}{-46}&\color{orangered}{-250} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -11x-46 } $ with a remainder of $ \color{red}{ -250 } $.