Divide using synthetic division $$ ( 3x^{2}+19x+33 ) \div ( 3x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-7&3&19&33\\& & -21& \color{black}{14} \\ \hline &\color{blue}{3}&\color{blue}{-2}&\color{orangered}{47} \end{array} $$The solution is:
$$ \frac{ 3x^{2}+19x+33 }{ x+7 } = \color{blue}{3x-2} ~+~ \frac{ \color{red}{ 47 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-7}&3&19&33\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-7&\color{orangered}{ 3 }&19&33\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 3 } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&3&19&33\\& & \color{blue}{-21} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrr}-7&3&\color{orangered}{ 19 }&33\\& & \color{orangered}{-21} & \\ \hline &3&\color{orangered}{-2}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&3&19&33\\& & -21& \color{blue}{14} \\ \hline &3&\color{blue}{-2}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 33 } + \color{orangered}{ 14 } = \color{orangered}{ 47 } $
$$ \begin{array}{c|rrr}-7&3&19&\color{orangered}{ 33 }\\& & -21& \color{orangered}{14} \\ \hline &\color{blue}{3}&\color{blue}{-2}&\color{orangered}{47} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x-2 } $ with a remainder of $ \color{red}{ 47 } $.