Divide using synthetic division $$ ( 3x^{4}-16x^{3}+21x^{2}+4x-12 ) \div ( 3x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&3&-16&21&4&-12\\& & -6& 44& -130& \color{black}{252} \\ \hline &\color{blue}{3}&\color{blue}{-22}&\color{blue}{65}&\color{blue}{-126}&\color{orangered}{240} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-16x^{3}+21x^{2}+4x-12 }{ x+2 } = \color{blue}{3x^{3}-22x^{2}+65x-126} ~+~ \frac{ \color{red}{ 240 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&3&-16&21&4&-12\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 3 }&-16&21&4&-12\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&3&-16&21&4&-12\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -22 } $
$$ \begin{array}{c|rrrrr}-2&3&\color{orangered}{ -16 }&21&4&-12\\& & \color{orangered}{-6} & & & \\ \hline &3&\color{orangered}{-22}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -22 \right) } = \color{blue}{ 44 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&3&-16&21&4&-12\\& & -6& \color{blue}{44} & & \\ \hline &3&\color{blue}{-22}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ 44 } = \color{orangered}{ 65 } $
$$ \begin{array}{c|rrrrr}-2&3&-16&\color{orangered}{ 21 }&4&-12\\& & -6& \color{orangered}{44} & & \\ \hline &3&-22&\color{orangered}{65}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 65 } = \color{blue}{ -130 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&3&-16&21&4&-12\\& & -6& 44& \color{blue}{-130} & \\ \hline &3&-22&\color{blue}{65}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -130 \right) } = \color{orangered}{ -126 } $
$$ \begin{array}{c|rrrrr}-2&3&-16&21&\color{orangered}{ 4 }&-12\\& & -6& 44& \color{orangered}{-130} & \\ \hline &3&-22&65&\color{orangered}{-126}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -126 \right) } = \color{blue}{ 252 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&3&-16&21&4&-12\\& & -6& 44& -130& \color{blue}{252} \\ \hline &3&-22&65&\color{blue}{-126}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 252 } = \color{orangered}{ 240 } $
$$ \begin{array}{c|rrrrr}-2&3&-16&21&4&\color{orangered}{ -12 }\\& & -6& 44& -130& \color{orangered}{252} \\ \hline &\color{blue}{3}&\color{blue}{-22}&\color{blue}{65}&\color{blue}{-126}&\color{orangered}{240} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-22x^{2}+65x-126 } $ with a remainder of $ \color{red}{ 240 } $.