Divide using synthetic division $$ ( 4x^{4}+2x^{3}-24x^{2}-18x+12 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&4&2&-24&-18&12\\& & -8& 12& 24& \color{black}{-12} \\ \hline &\color{blue}{4}&\color{blue}{-6}&\color{blue}{-12}&\color{blue}{6}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 4x^{4}+2x^{3}-24x^{2}-18x+12 }{ x+2 } = \color{blue}{4x^{3}-6x^{2}-12x+6} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&2&-24&-18&12\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 4 }&2&-24&-18&12\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&2&-24&-18&12\\& & \color{blue}{-8} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}-2&4&\color{orangered}{ 2 }&-24&-18&12\\& & \color{orangered}{-8} & & & \\ \hline &4&\color{orangered}{-6}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&2&-24&-18&12\\& & -8& \color{blue}{12} & & \\ \hline &4&\color{blue}{-6}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 12 } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrr}-2&4&2&\color{orangered}{ -24 }&-18&12\\& & -8& \color{orangered}{12} & & \\ \hline &4&-6&\color{orangered}{-12}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&2&-24&-18&12\\& & -8& 12& \color{blue}{24} & \\ \hline &4&-6&\color{blue}{-12}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 24 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}-2&4&2&-24&\color{orangered}{ -18 }&12\\& & -8& 12& \color{orangered}{24} & \\ \hline &4&-6&-12&\color{orangered}{6}& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&4&2&-24&-18&12\\& & -8& 12& 24& \color{blue}{-12} \\ \hline &4&-6&-12&\color{blue}{6}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-2&4&2&-24&-18&\color{orangered}{ 12 }\\& & -8& 12& 24& \color{orangered}{-12} \\ \hline &\color{blue}{4}&\color{blue}{-6}&\color{blue}{-12}&\color{blue}{6}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}-6x^{2}-12x+6 } $ with a remainder of $ \color{red}{ 0 } $.