Divide using synthetic division $$ ( 6x^{2}-12 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&6&0&-12\\& & -12& \color{black}{24} \\ \hline &\color{blue}{6}&\color{blue}{-12}&\color{orangered}{12} \end{array} $$The solution is:
$$ \frac{ 6x^{2}-12 }{ x+2 } = \color{blue}{6x-12} ~+~ \frac{ \color{red}{ 12 } }{ x+2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&6&0&-12\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 6 }&0&-12\\& & & \\ \hline &\color{orangered}{6}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&6&0&-12\\& & \color{blue}{-12} & \\ \hline &\color{blue}{6}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrr}-2&6&\color{orangered}{ 0 }&-12\\& & \color{orangered}{-12} & \\ \hline &6&\color{orangered}{-12}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&6&0&-12\\& & -12& \color{blue}{24} \\ \hline &6&\color{blue}{-12}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 24 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrr}-2&6&0&\color{orangered}{ -12 }\\& & -12& \color{orangered}{24} \\ \hline &\color{blue}{6}&\color{blue}{-12}&\color{orangered}{12} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x-12 } $ with a remainder of $ \color{red}{ 12 } $.