Divide using synthetic division $$ ( -11x^{2}+13x-44 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&-11&13&-44\\& & -55& \color{black}{-210} \\ \hline &\color{blue}{-11}&\color{blue}{-42}&\color{orangered}{-254} \end{array} $$The solution is:
$$ \frac{ -11x^{2}+13x-44 }{ x-5 } = \color{blue}{-11x-42} \color{red}{~-~} \frac{ \color{red}{ 254 } }{ x-5 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{5}&-11&13&-44\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ -11 }&13&-44\\& & & \\ \hline &\color{orangered}{-11}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-11&13&-44\\& & \color{blue}{-55} & \\ \hline &\color{blue}{-11}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -55 \right) } = \color{orangered}{ -42 } $
$$ \begin{array}{c|rrr}5&-11&\color{orangered}{ 13 }&-44\\& & \color{orangered}{-55} & \\ \hline &-11&\color{orangered}{-42}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -42 \right) } = \color{blue}{ -210 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-11&13&-44\\& & -55& \color{blue}{-210} \\ \hline &-11&\color{blue}{-42}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -44 } + \color{orangered}{ \left( -210 \right) } = \color{orangered}{ -254 } $
$$ \begin{array}{c|rrr}5&-11&13&\color{orangered}{ -44 }\\& & -55& \color{orangered}{-210} \\ \hline &\color{blue}{-11}&\color{blue}{-42}&\color{orangered}{-254} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -11x-42 } $ with a remainder of $ \color{red}{ -254 } $.