Divide using synthetic division $$ ( 6x^{2}+23x+23 ) \div ( 3x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-7&6&23&23\\& & -42& \color{black}{133} \\ \hline &\color{blue}{6}&\color{blue}{-19}&\color{orangered}{156} \end{array} $$The solution is:
$$ \frac{ 6x^{2}+23x+23 }{ x+7 } = \color{blue}{6x-19} ~+~ \frac{ \color{red}{ 156 } }{ x+7 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-7}&6&23&23\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-7&\color{orangered}{ 6 }&23&23\\& & & \\ \hline &\color{orangered}{6}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 6 } = \color{blue}{ -42 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&6&23&23\\& & \color{blue}{-42} & \\ \hline &\color{blue}{6}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrr}-7&6&\color{orangered}{ 23 }&23\\& & \color{orangered}{-42} & \\ \hline &6&\color{orangered}{-19}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ 133 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&6&23&23\\& & -42& \color{blue}{133} \\ \hline &6&\color{blue}{-19}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ 133 } = \color{orangered}{ 156 } $
$$ \begin{array}{c|rrr}-7&6&23&\color{orangered}{ 23 }\\& & -42& \color{orangered}{133} \\ \hline &\color{blue}{6}&\color{blue}{-19}&\color{orangered}{156} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x-19 } $ with a remainder of $ \color{red}{ 156 } $.