Divide using synthetic division $$ ( -5x^{2}-33x-37 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}9&-5&-33&-37\\& & -45& \color{black}{-702} \\ \hline &\color{blue}{-5}&\color{blue}{-78}&\color{orangered}{-739} \end{array} $$The solution is:
$$ \frac{ -5x^{2}-33x-37 }{ x-9 } = \color{blue}{-5x-78} \color{red}{~-~} \frac{ \color{red}{ 739 } }{ x-9 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{9}&-5&-33&-37\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}9&\color{orangered}{ -5 }&-33&-37\\& & & \\ \hline &\color{orangered}{-5}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -45 } $.
$$ \begin{array}{c|rrr}\color{blue}{9}&-5&-33&-37\\& & \color{blue}{-45} & \\ \hline &\color{blue}{-5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -33 } + \color{orangered}{ \left( -45 \right) } = \color{orangered}{ -78 } $
$$ \begin{array}{c|rrr}9&-5&\color{orangered}{ -33 }&-37\\& & \color{orangered}{-45} & \\ \hline &-5&\color{orangered}{-78}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -78 \right) } = \color{blue}{ -702 } $.
$$ \begin{array}{c|rrr}\color{blue}{9}&-5&-33&-37\\& & -45& \color{blue}{-702} \\ \hline &-5&\color{blue}{-78}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -37 } + \color{orangered}{ \left( -702 \right) } = \color{orangered}{ -739 } $
$$ \begin{array}{c|rrr}9&-5&-33&\color{orangered}{ -37 }\\& & -45& \color{orangered}{-702} \\ \hline &\color{blue}{-5}&\color{blue}{-78}&\color{orangered}{-739} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x-78 } $ with a remainder of $ \color{red}{ -739 } $.