Divide using synthetic division $$ ( -3x^{3}+4x^{2}+9x+11 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&-3&4&9&11\\& & -9& -15& \color{black}{-18} \\ \hline &\color{blue}{-3}&\color{blue}{-5}&\color{blue}{-6}&\color{orangered}{-7} \end{array} $$The solution is:
$$ \frac{ -3x^{3}+4x^{2}+9x+11 }{ x-3 } = \color{blue}{-3x^{2}-5x-6} \color{red}{~-~} \frac{ \color{red}{ 7 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&4&9&11\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ -3 }&4&9&11\\& & & & \\ \hline &\color{orangered}{-3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&4&9&11\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{-3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}3&-3&\color{orangered}{ 4 }&9&11\\& & \color{orangered}{-9} & & \\ \hline &-3&\color{orangered}{-5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&4&9&11\\& & -9& \color{blue}{-15} & \\ \hline &-3&\color{blue}{-5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}3&-3&4&\color{orangered}{ 9 }&11\\& & -9& \color{orangered}{-15} & \\ \hline &-3&-5&\color{orangered}{-6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&4&9&11\\& & -9& -15& \color{blue}{-18} \\ \hline &-3&-5&\color{blue}{-6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}3&-3&4&9&\color{orangered}{ 11 }\\& & -9& -15& \color{orangered}{-18} \\ \hline &\color{blue}{-3}&\color{blue}{-5}&\color{blue}{-6}&\color{orangered}{-7} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{2}-5x-6 } $ with a remainder of $ \color{red}{ -7 } $.