Divide using synthetic division $$ ( 3x^{3}-9x^{2}-4x+12 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&3&-9&-4&12\\& & -6& 30& \color{black}{-52} \\ \hline &\color{blue}{3}&\color{blue}{-15}&\color{blue}{26}&\color{orangered}{-40} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-9x^{2}-4x+12 }{ x+2 } = \color{blue}{3x^{2}-15x+26} \color{red}{~-~} \frac{ \color{red}{ 40 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-9&-4&12\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 3 }&-9&-4&12\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-9&-4&12\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrr}-2&3&\color{orangered}{ -9 }&-4&12\\& & \color{orangered}{-6} & & \\ \hline &3&\color{orangered}{-15}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-9&-4&12\\& & -6& \color{blue}{30} & \\ \hline &3&\color{blue}{-15}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 30 } = \color{orangered}{ 26 } $
$$ \begin{array}{c|rrrr}-2&3&-9&\color{orangered}{ -4 }&12\\& & -6& \color{orangered}{30} & \\ \hline &3&-15&\color{orangered}{26}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 26 } = \color{blue}{ -52 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-9&-4&12\\& & -6& 30& \color{blue}{-52} \\ \hline &3&-15&\color{blue}{26}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -52 \right) } = \color{orangered}{ -40 } $
$$ \begin{array}{c|rrrr}-2&3&-9&-4&\color{orangered}{ 12 }\\& & -6& 30& \color{orangered}{-52} \\ \hline &\color{blue}{3}&\color{blue}{-15}&\color{blue}{26}&\color{orangered}{-40} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-15x+26 } $ with a remainder of $ \color{red}{ -40 } $.