Divide using synthetic division $$ ( 12x^{3}-6x^{2}+16x ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&12&-6&16&0\\& & -24& 60& \color{black}{-152} \\ \hline &\color{blue}{12}&\color{blue}{-30}&\color{blue}{76}&\color{orangered}{-152} \end{array} $$The solution is:
$$ \frac{ 12x^{3}-6x^{2}+16x }{ x+2 } = \color{blue}{12x^{2}-30x+76} \color{red}{~-~} \frac{ \color{red}{ 152 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&-6&16&0\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 12 }&-6&16&0\\& & & & \\ \hline &\color{orangered}{12}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 12 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&-6&16&0\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{12}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -30 } $
$$ \begin{array}{c|rrrr}-2&12&\color{orangered}{ -6 }&16&0\\& & \color{orangered}{-24} & & \\ \hline &12&\color{orangered}{-30}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -30 \right) } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&-6&16&0\\& & -24& \color{blue}{60} & \\ \hline &12&\color{blue}{-30}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ 60 } = \color{orangered}{ 76 } $
$$ \begin{array}{c|rrrr}-2&12&-6&\color{orangered}{ 16 }&0\\& & -24& \color{orangered}{60} & \\ \hline &12&-30&\color{orangered}{76}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 76 } = \color{blue}{ -152 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&12&-6&16&0\\& & -24& 60& \color{blue}{-152} \\ \hline &12&-30&\color{blue}{76}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -152 \right) } = \color{orangered}{ -152 } $
$$ \begin{array}{c|rrrr}-2&12&-6&16&\color{orangered}{ 0 }\\& & -24& 60& \color{orangered}{-152} \\ \hline &\color{blue}{12}&\color{blue}{-30}&\color{blue}{76}&\color{orangered}{-152} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 12x^{2}-30x+76 } $ with a remainder of $ \color{red}{ -152 } $.