Divide using synthetic division $$ ( 9x^{2}+8 ) \div ( 3x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&9&0&8\\& & -18& \color{black}{36} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{orangered}{44} \end{array} $$The solution is:
$$ \frac{ 9x^{2}+8 }{ x+2 } = \color{blue}{9x-18} ~+~ \frac{ \color{red}{ 44 } }{ x+2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&9&0&8\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 9 }&0&8\\& & & \\ \hline &\color{orangered}{9}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&9&0&8\\& & \color{blue}{-18} & \\ \hline &\color{blue}{9}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrr}-2&9&\color{orangered}{ 0 }&8\\& & \color{orangered}{-18} & \\ \hline &9&\color{orangered}{-18}& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&9&0&8\\& & -18& \color{blue}{36} \\ \hline &9&\color{blue}{-18}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 36 } = \color{orangered}{ 44 } $
$$ \begin{array}{c|rrr}-2&9&0&\color{orangered}{ 8 }\\& & -18& \color{orangered}{36} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{orangered}{44} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x-18 } $ with a remainder of $ \color{red}{ 44 } $.