Divide using synthetic division $$ ( 3x^{3}-3x^{2}-4 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&3&-3&0&-4\\& & -6& 18& \color{black}{-36} \\ \hline &\color{blue}{3}&\color{blue}{-9}&\color{blue}{18}&\color{orangered}{-40} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-3x^{2}-4 }{ x+2 } = \color{blue}{3x^{2}-9x+18} \color{red}{~-~} \frac{ \color{red}{ 40 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-3&0&-4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 3 }&-3&0&-4\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-3&0&-4\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}-2&3&\color{orangered}{ -3 }&0&-4\\& & \color{orangered}{-6} & & \\ \hline &3&\color{orangered}{-9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-3&0&-4\\& & -6& \color{blue}{18} & \\ \hline &3&\color{blue}{-9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}-2&3&-3&\color{orangered}{ 0 }&-4\\& & -6& \color{orangered}{18} & \\ \hline &3&-9&\color{orangered}{18}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 18 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-3&0&-4\\& & -6& 18& \color{blue}{-36} \\ \hline &3&-9&\color{blue}{18}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -40 } $
$$ \begin{array}{c|rrrr}-2&3&-3&0&\color{orangered}{ -4 }\\& & -6& 18& \color{orangered}{-36} \\ \hline &\color{blue}{3}&\color{blue}{-9}&\color{blue}{18}&\color{orangered}{-40} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-9x+18 } $ with a remainder of $ \color{red}{ -40 } $.