Divide using synthetic division $$ ( 2x^{2}-2x+1 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&2&-2&1\\& & -4& \color{black}{12} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{orangered}{13} \end{array} $$The solution is:
$$ \frac{ 2x^{2}-2x+1 }{ x+2 } = \color{blue}{2x-6} ~+~ \frac{ \color{red}{ 13 } }{ x+2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-2&1\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 2 }&-2&1\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-2&1\\& & \color{blue}{-4} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrr}-2&2&\color{orangered}{ -2 }&1\\& & \color{orangered}{-4} & \\ \hline &2&\color{orangered}{-6}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-2&1\\& & -4& \color{blue}{12} \\ \hline &2&\color{blue}{-6}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 12 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrr}-2&2&-2&\color{orangered}{ 1 }\\& & -4& \color{orangered}{12} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{orangered}{13} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x-6 } $ with a remainder of $ \color{red}{ 13 } $.