Divide using synthetic division $$ ( -2x^{3}+7x^{2}+33 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&-2&7&0&33\\& & -12& -30& \color{black}{-180} \\ \hline &\color{blue}{-2}&\color{blue}{-5}&\color{blue}{-30}&\color{orangered}{-147} \end{array} $$The solution is:
$$ \frac{ -2x^{3}+7x^{2}+33 }{ x-6 } = \color{blue}{-2x^{2}-5x-30} \color{red}{~-~} \frac{ \color{red}{ 147 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{6}&-2&7&0&33\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}6&\color{orangered}{ -2 }&7&0&33\\& & & & \\ \hline &\color{orangered}{-2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&-2&7&0&33\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{-2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}6&-2&\color{orangered}{ 7 }&0&33\\& & \color{orangered}{-12} & & \\ \hline &-2&\color{orangered}{-5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&-2&7&0&33\\& & -12& \color{blue}{-30} & \\ \hline &-2&\color{blue}{-5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -30 } $
$$ \begin{array}{c|rrrr}6&-2&7&\color{orangered}{ 0 }&33\\& & -12& \color{orangered}{-30} & \\ \hline &-2&-5&\color{orangered}{-30}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -30 \right) } = \color{blue}{ -180 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&-2&7&0&33\\& & -12& -30& \color{blue}{-180} \\ \hline &-2&-5&\color{blue}{-30}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 33 } + \color{orangered}{ \left( -180 \right) } = \color{orangered}{ -147 } $
$$ \begin{array}{c|rrrr}6&-2&7&0&\color{orangered}{ 33 }\\& & -12& -30& \color{orangered}{-180} \\ \hline &\color{blue}{-2}&\color{blue}{-5}&\color{blue}{-30}&\color{orangered}{-147} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{2}-5x-30 } $ with a remainder of $ \color{red}{ -147 } $.