Divide using synthetic division $$ ( 3x^{3}+2x^{2}-3x-2 ) \div ( 3x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&3&2&-3&-2\\& & -6& 8& \color{black}{-10} \\ \hline &\color{blue}{3}&\color{blue}{-4}&\color{blue}{5}&\color{orangered}{-12} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+2x^{2}-3x-2 }{ x+2 } = \color{blue}{3x^{2}-4x+5} \color{red}{~-~} \frac{ \color{red}{ 12 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&2&-3&-2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 3 }&2&-3&-2\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&2&-3&-2\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-2&3&\color{orangered}{ 2 }&-3&-2\\& & \color{orangered}{-6} & & \\ \hline &3&\color{orangered}{-4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&2&-3&-2\\& & -6& \color{blue}{8} & \\ \hline &3&\color{blue}{-4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 8 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}-2&3&2&\color{orangered}{ -3 }&-2\\& & -6& \color{orangered}{8} & \\ \hline &3&-4&\color{orangered}{5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&2&-3&-2\\& & -6& 8& \color{blue}{-10} \\ \hline &3&-4&\color{blue}{5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrr}-2&3&2&-3&\color{orangered}{ -2 }\\& & -6& 8& \color{orangered}{-10} \\ \hline &\color{blue}{3}&\color{blue}{-4}&\color{blue}{5}&\color{orangered}{-12} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-4x+5 } $ with a remainder of $ \color{red}{ -12 } $.