Divide using synthetic division $$ ( 6x^{4}+10x^{3}-2x^{2}+8x+8 ) \div ( 3x+12 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-12&6&10&-2&8&8\\& & -72& 744& -8904& \color{black}{106752} \\ \hline &\color{blue}{6}&\color{blue}{-62}&\color{blue}{742}&\color{blue}{-8896}&\color{orangered}{106760} \end{array} $$The solution is:
$$ \frac{ 6x^{4}+10x^{3}-2x^{2}+8x+8 }{ x+12 } = \color{blue}{6x^{3}-62x^{2}+742x-8896} ~+~ \frac{ \color{red}{ 106760 } }{ x+12 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 12 = 0 $ ( $ x = \color{blue}{ -12 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-12}&6&10&-2&8&8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-12&\color{orangered}{ 6 }&10&-2&8&8\\& & & & & \\ \hline &\color{orangered}{6}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ 6 } = \color{blue}{ -72 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-12}&6&10&-2&8&8\\& & \color{blue}{-72} & & & \\ \hline &\color{blue}{6}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -72 \right) } = \color{orangered}{ -62 } $
$$ \begin{array}{c|rrrrr}-12&6&\color{orangered}{ 10 }&-2&8&8\\& & \color{orangered}{-72} & & & \\ \hline &6&\color{orangered}{-62}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ \left( -62 \right) } = \color{blue}{ 744 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-12}&6&10&-2&8&8\\& & -72& \color{blue}{744} & & \\ \hline &6&\color{blue}{-62}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 744 } = \color{orangered}{ 742 } $
$$ \begin{array}{c|rrrrr}-12&6&10&\color{orangered}{ -2 }&8&8\\& & -72& \color{orangered}{744} & & \\ \hline &6&-62&\color{orangered}{742}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ 742 } = \color{blue}{ -8904 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-12}&6&10&-2&8&8\\& & -72& 744& \color{blue}{-8904} & \\ \hline &6&-62&\color{blue}{742}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -8904 \right) } = \color{orangered}{ -8896 } $
$$ \begin{array}{c|rrrrr}-12&6&10&-2&\color{orangered}{ 8 }&8\\& & -72& 744& \color{orangered}{-8904} & \\ \hline &6&-62&742&\color{orangered}{-8896}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ \left( -8896 \right) } = \color{blue}{ 106752 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-12}&6&10&-2&8&8\\& & -72& 744& -8904& \color{blue}{106752} \\ \hline &6&-62&742&\color{blue}{-8896}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 106752 } = \color{orangered}{ 106760 } $
$$ \begin{array}{c|rrrrr}-12&6&10&-2&8&\color{orangered}{ 8 }\\& & -72& 744& -8904& \color{orangered}{106752} \\ \hline &\color{blue}{6}&\color{blue}{-62}&\color{blue}{742}&\color{blue}{-8896}&\color{orangered}{106760} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{3}-62x^{2}+742x-8896 } $ with a remainder of $ \color{red}{ 106760 } $.