Divide using synthetic division $$ ( x^{4}-8x^{3}+12x^{2}+6x-6 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&1&-8&12&6&-6\\& & 2& -12& 0& \color{black}{12} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{0}&\color{blue}{6}&\color{orangered}{6} \end{array} $$The solution is:
$$ \frac{ x^{4}-8x^{3}+12x^{2}+6x-6 }{ x-2 } = \color{blue}{x^{3}-6x^{2}+6} ~+~ \frac{ \color{red}{ 6 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-8&12&6&-6\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ 1 }&-8&12&6&-6\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-8&12&6&-6\\& & \color{blue}{2} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 2 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}2&1&\color{orangered}{ -8 }&12&6&-6\\& & \color{orangered}{2} & & & \\ \hline &1&\color{orangered}{-6}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-8&12&6&-6\\& & 2& \color{blue}{-12} & & \\ \hline &1&\color{blue}{-6}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}2&1&-8&\color{orangered}{ 12 }&6&-6\\& & 2& \color{orangered}{-12} & & \\ \hline &1&-6&\color{orangered}{0}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-8&12&6&-6\\& & 2& -12& \color{blue}{0} & \\ \hline &1&-6&\color{blue}{0}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 0 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}2&1&-8&12&\color{orangered}{ 6 }&-6\\& & 2& -12& \color{orangered}{0} & \\ \hline &1&-6&0&\color{orangered}{6}& \end{array} $$Step 8 : 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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&1&-8&12&6&-6\\& & 2& -12& 0& \color{blue}{12} \\ \hline &1&-6&0&\color{blue}{6}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 12 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}2&1&-8&12&6&\color{orangered}{ -6 }\\& & 2& -12& 0& \color{orangered}{12} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{0}&\color{blue}{6}&\color{orangered}{6} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-6x^{2}+6 } $ with a remainder of $ \color{red}{ 6 } $.