Divide using synthetic division $$ ( 6x^{3}+20x^{2}+23x+21 ) \div ( 3x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-7&6&20&23&21\\& & -42& 154& \color{black}{-1239} \\ \hline &\color{blue}{6}&\color{blue}{-22}&\color{blue}{177}&\color{orangered}{-1218} \end{array} $$The solution is:
$$ \frac{ 6x^{3}+20x^{2}+23x+21 }{ x+7 } = \color{blue}{6x^{2}-22x+177} \color{red}{~-~} \frac{ \color{red}{ 1218 } }{ x+7 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&6&20&23&21\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-7&\color{orangered}{ 6 }&20&23&21\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&6&20&23&21\\& & \color{blue}{-42} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ -22 } $
$$ \begin{array}{c|rrrr}-7&6&\color{orangered}{ 20 }&23&21\\& & \color{orangered}{-42} & & \\ \hline &6&\color{orangered}{-22}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -22 \right) } = \color{blue}{ 154 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&6&20&23&21\\& & -42& \color{blue}{154} & \\ \hline &6&\color{blue}{-22}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ 154 } = \color{orangered}{ 177 } $
$$ \begin{array}{c|rrrr}-7&6&20&\color{orangered}{ 23 }&21\\& & -42& \color{orangered}{154} & \\ \hline &6&-22&\color{orangered}{177}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 177 } = \color{blue}{ -1239 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&6&20&23&21\\& & -42& 154& \color{blue}{-1239} \\ \hline &6&-22&\color{blue}{177}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -1239 \right) } = \color{orangered}{ -1218 } $
$$ \begin{array}{c|rrrr}-7&6&20&23&\color{orangered}{ 21 }\\& & -42& 154& \color{orangered}{-1239} \\ \hline &\color{blue}{6}&\color{blue}{-22}&\color{blue}{177}&\color{orangered}{-1218} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}-22x+177 } $ with a remainder of $ \color{red}{ -1218 } $.