Divide using synthetic division $$ ( -2x^{3}+5x^{2}+9 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&-2&5&0&9\\& & 6& -33& \color{black}{99} \\ \hline &\color{blue}{-2}&\color{blue}{11}&\color{blue}{-33}&\color{orangered}{108} \end{array} $$The solution is:
$$ \frac{ -2x^{3}+5x^{2}+9 }{ x+3 } = \color{blue}{-2x^{2}+11x-33} ~+~ \frac{ \color{red}{ 108 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&-2&5&0&9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ -2 }&5&0&9\\& & & & \\ \hline &\color{orangered}{-2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&-2&5&0&9\\& & \color{blue}{6} & & \\ \hline &\color{blue}{-2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 6 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrr}-3&-2&\color{orangered}{ 5 }&0&9\\& & \color{orangered}{6} & & \\ \hline &-2&\color{orangered}{11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 11 } = \color{blue}{ -33 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&-2&5&0&9\\& & 6& \color{blue}{-33} & \\ \hline &-2&\color{blue}{11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -33 \right) } = \color{orangered}{ -33 } $
$$ \begin{array}{c|rrrr}-3&-2&5&\color{orangered}{ 0 }&9\\& & 6& \color{orangered}{-33} & \\ \hline &-2&11&\color{orangered}{-33}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -33 \right) } = \color{blue}{ 99 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&-2&5&0&9\\& & 6& -33& \color{blue}{99} \\ \hline &-2&11&\color{blue}{-33}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 99 } = \color{orangered}{ 108 } $
$$ \begin{array}{c|rrrr}-3&-2&5&0&\color{orangered}{ 9 }\\& & 6& -33& \color{orangered}{99} \\ \hline &\color{blue}{-2}&\color{blue}{11}&\color{blue}{-33}&\color{orangered}{108} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{2}+11x-33 } $ with a remainder of $ \color{red}{ 108 } $.