Divide using synthetic division $$ ( 3x^{2}+4 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&3&0&4\\& & -6& \color{black}{12} \\ \hline &\color{blue}{3}&\color{blue}{-6}&\color{orangered}{16} \end{array} $$The solution is:
$$ \frac{ 3x^{2}+4 }{ x+2 } = \color{blue}{3x-6} ~+~ \frac{ \color{red}{ 16 } }{ x+2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&3&0&4\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 3 }&0&4\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&3&0&4\\& & \color{blue}{-6} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrr}-2&3&\color{orangered}{ 0 }&4\\& & \color{orangered}{-6} & \\ \hline &3&\color{orangered}{-6}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&3&0&4\\& & -6& \color{blue}{12} \\ \hline &3&\color{blue}{-6}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 12 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrr}-2&3&0&\color{orangered}{ 4 }\\& & -6& \color{orangered}{12} \\ \hline &\color{blue}{3}&\color{blue}{-6}&\color{orangered}{16} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x-6 } $ with a remainder of $ \color{red}{ 16 } $.