Divide using synthetic division $$ ( 9x^{2}+15x+6 ) \div ( 3x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&9&15&6\\& & -18& \color{black}{6} \\ \hline &\color{blue}{9}&\color{blue}{-3}&\color{orangered}{12} \end{array} $$The solution is:
$$ \frac{ 9x^{2}+15x+6 }{ x+2 } = \color{blue}{9x-3} ~+~ \frac{ \color{red}{ 12 } }{ x+2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&9&15&6\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 9 }&15&6\\& & & \\ \hline &\color{orangered}{9}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 9 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&9&15&6\\& & \color{blue}{-18} & \\ \hline &\color{blue}{9}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrr}-2&9&\color{orangered}{ 15 }&6\\& & \color{orangered}{-18} & \\ \hline &9&\color{orangered}{-3}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&9&15&6\\& & -18& \color{blue}{6} \\ \hline &9&\color{blue}{-3}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 6 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrr}-2&9&15&\color{orangered}{ 6 }\\& & -18& \color{orangered}{6} \\ \hline &\color{blue}{9}&\color{blue}{-3}&\color{orangered}{12} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x-3 } $ with a remainder of $ \color{red}{ 12 } $.