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