Divide using synthetic division $$ ( 9x^{3}+2x+15 ) \div ( 3x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&9&0&2&15\\& & -18& 36& \color{black}{-76} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{blue}{38}&\color{orangered}{-61} \end{array} $$The solution is:
$$ \frac{ 9x^{3}+2x+15 }{ x+2 } = \color{blue}{9x^{2}-18x+38} \color{red}{~-~} \frac{ \color{red}{ 61 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&9&0&2&15\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 9 }&0&2&15\\& & & & \\ \hline &\color{orangered}{9}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&9&0&2&15\\& & \color{blue}{-18} & & \\ \hline &\color{blue}{9}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrr}-2&9&\color{orangered}{ 0 }&2&15\\& & \color{orangered}{-18} & & \\ \hline &9&\color{orangered}{-18}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&9&0&2&15\\& & -18& \color{blue}{36} & \\ \hline &9&\color{blue}{-18}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 36 } = \color{orangered}{ 38 } $
$$ \begin{array}{c|rrrr}-2&9&0&\color{orangered}{ 2 }&15\\& & -18& \color{orangered}{36} & \\ \hline &9&-18&\color{orangered}{38}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 38 } = \color{blue}{ -76 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&9&0&2&15\\& & -18& 36& \color{blue}{-76} \\ \hline &9&-18&\color{blue}{38}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -76 \right) } = \color{orangered}{ -61 } $
$$ \begin{array}{c|rrrr}-2&9&0&2&\color{orangered}{ 15 }\\& & -18& 36& \color{orangered}{-76} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{blue}{38}&\color{orangered}{-61} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{2}-18x+38 } $ with a remainder of $ \color{red}{ -61 } $.