Divide using synthetic division $$ ( 6x^{3}+23x^{2}+10x+1 ) \div ( 3x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&6&23&10&1\\& & -6& -17& \color{black}{7} \\ \hline &\color{blue}{6}&\color{blue}{17}&\color{blue}{-7}&\color{orangered}{8} \end{array} $$The solution is:
$$ \frac{ 6x^{3}+23x^{2}+10x+1 }{ x+1 } = \color{blue}{6x^{2}+17x-7} ~+~ \frac{ \color{red}{ 8 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&6&23&10&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 6 }&23&10&1\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 6 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&6&23&10&1\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrr}-1&6&\color{orangered}{ 23 }&10&1\\& & \color{orangered}{-6} & & \\ \hline &6&\color{orangered}{17}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 17 } = \color{blue}{ -17 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&6&23&10&1\\& & -6& \color{blue}{-17} & \\ \hline &6&\color{blue}{17}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -17 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}-1&6&23&\color{orangered}{ 10 }&1\\& & -6& \color{orangered}{-17} & \\ \hline &6&17&\color{orangered}{-7}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 7 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&6&23&10&1\\& & -6& -17& \color{blue}{7} \\ \hline &6&17&\color{blue}{-7}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 7 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}-1&6&23&10&\color{orangered}{ 1 }\\& & -6& -17& \color{orangered}{7} \\ \hline &\color{blue}{6}&\color{blue}{17}&\color{blue}{-7}&\color{orangered}{8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+17x-7 } $ with a remainder of $ \color{red}{ 8 } $.