Divide using synthetic division $$ ( 6x^{2}-2x+2 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&6&-2&2\\& & -12& \color{black}{28} \\ \hline &\color{blue}{6}&\color{blue}{-14}&\color{orangered}{30} \end{array} $$The solution is:
$$ \frac{ 6x^{2}-2x+2 }{ x+2 } = \color{blue}{6x-14} ~+~ \frac{ \color{red}{ 30 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-2}&6&-2&2\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 6 }&-2&2\\& & & \\ \hline &\color{orangered}{6}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 6 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&6&-2&2\\& & \color{blue}{-12} & \\ \hline &\color{blue}{6}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrr}-2&6&\color{orangered}{ -2 }&2\\& & \color{orangered}{-12} & \\ \hline &6&\color{orangered}{-14}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&6&-2&2\\& & -12& \color{blue}{28} \\ \hline &6&\color{blue}{-14}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 28 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrr}-2&6&-2&\color{orangered}{ 2 }\\& & -12& \color{orangered}{28} \\ \hline &\color{blue}{6}&\color{blue}{-14}&\color{orangered}{30} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x-14 } $ with a remainder of $ \color{red}{ 30 } $.