Divide using synthetic division $$ ( 2x^{2}+4x+6 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}1&2&4&6\\& & 2& \color{black}{6} \\ \hline &\color{blue}{2}&\color{blue}{6}&\color{orangered}{12} \end{array} $$The solution is:
$$ \frac{ 2x^{2}+4x+6 }{ x-1 } = \color{blue}{2x+6} ~+~ \frac{ \color{red}{ 12 } }{ x-1 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{1}&2&4&6\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}1&\color{orangered}{ 2 }&4&6\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 2 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&2&4&6\\& & \color{blue}{2} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 2 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrr}1&2&\color{orangered}{ 4 }&6\\& & \color{orangered}{2} & \\ \hline &2&\color{orangered}{6}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 6 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&2&4&6\\& & 2& \color{blue}{6} \\ \hline &2&\color{blue}{6}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 6 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrr}1&2&4&\color{orangered}{ 6 }\\& & 2& \color{orangered}{6} \\ \hline &\color{blue}{2}&\color{blue}{6}&\color{orangered}{12} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x+6 } $ with a remainder of $ \color{red}{ 12 } $.