Divide using synthetic division $$ ( 4x^{4}-8x^{3}+12x^{2}-6x+12 ) \div ( 2x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&4&-8&12&-6&12\\& & -16& 96& -432& \color{black}{1752} \\ \hline &\color{blue}{4}&\color{blue}{-24}&\color{blue}{108}&\color{blue}{-438}&\color{orangered}{1764} \end{array} $$The solution is:
$$ \frac{ 4x^{4}-8x^{3}+12x^{2}-6x+12 }{ x+4 } = \color{blue}{4x^{3}-24x^{2}+108x-438} ~+~ \frac{ \color{red}{ 1764 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&4&-8&12&-6&12\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 4 }&-8&12&-6&12\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&4&-8&12&-6&12\\& & \color{blue}{-16} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrrr}-4&4&\color{orangered}{ -8 }&12&-6&12\\& & \color{orangered}{-16} & & & \\ \hline &4&\color{orangered}{-24}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ 96 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&4&-8&12&-6&12\\& & -16& \color{blue}{96} & & \\ \hline &4&\color{blue}{-24}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ 96 } = \color{orangered}{ 108 } $
$$ \begin{array}{c|rrrrr}-4&4&-8&\color{orangered}{ 12 }&-6&12\\& & -16& \color{orangered}{96} & & \\ \hline &4&-24&\color{orangered}{108}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 108 } = \color{blue}{ -432 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&4&-8&12&-6&12\\& & -16& 96& \color{blue}{-432} & \\ \hline &4&-24&\color{blue}{108}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -432 \right) } = \color{orangered}{ -438 } $
$$ \begin{array}{c|rrrrr}-4&4&-8&12&\color{orangered}{ -6 }&12\\& & -16& 96& \color{orangered}{-432} & \\ \hline &4&-24&108&\color{orangered}{-438}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -438 \right) } = \color{blue}{ 1752 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&4&-8&12&-6&12\\& & -16& 96& -432& \color{blue}{1752} \\ \hline &4&-24&108&\color{blue}{-438}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ 1752 } = \color{orangered}{ 1764 } $
$$ \begin{array}{c|rrrrr}-4&4&-8&12&-6&\color{orangered}{ 12 }\\& & -16& 96& -432& \color{orangered}{1752} \\ \hline &\color{blue}{4}&\color{blue}{-24}&\color{blue}{108}&\color{blue}{-438}&\color{orangered}{1764} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}-24x^{2}+108x-438 } $ with a remainder of $ \color{red}{ 1764 } $.