Divide using synthetic division $$ ( 9x^{3}-21x^{2}-6x-6 ) \div ( 3x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&9&-21&-6&-6\\& & -18& 78& \color{black}{-144} \\ \hline &\color{blue}{9}&\color{blue}{-39}&\color{blue}{72}&\color{orangered}{-150} \end{array} $$The solution is:
$$ \frac{ 9x^{3}-21x^{2}-6x-6 }{ x+2 } = \color{blue}{9x^{2}-39x+72} \color{red}{~-~} \frac{ \color{red}{ 150 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&9&-21&-6&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 9 }&-21&-6&-6\\& & & & \\ \hline &\color{orangered}{9}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 9 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&9&-21&-6&-6\\& & \color{blue}{-18} & & \\ \hline &\color{blue}{9}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -39 } $
$$ \begin{array}{c|rrrr}-2&9&\color{orangered}{ -21 }&-6&-6\\& & \color{orangered}{-18} & & \\ \hline &9&\color{orangered}{-39}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -39 \right) } = \color{blue}{ 78 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&9&-21&-6&-6\\& & -18& \color{blue}{78} & \\ \hline &9&\color{blue}{-39}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 78 } = \color{orangered}{ 72 } $
$$ \begin{array}{c|rrrr}-2&9&-21&\color{orangered}{ -6 }&-6\\& & -18& \color{orangered}{78} & \\ \hline &9&-39&\color{orangered}{72}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 72 } = \color{blue}{ -144 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&9&-21&-6&-6\\& & -18& 78& \color{blue}{-144} \\ \hline &9&-39&\color{blue}{72}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -144 \right) } = \color{orangered}{ -150 } $
$$ \begin{array}{c|rrrr}-2&9&-21&-6&\color{orangered}{ -6 }\\& & -18& 78& \color{orangered}{-144} \\ \hline &\color{blue}{9}&\color{blue}{-39}&\color{blue}{72}&\color{orangered}{-150} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{2}-39x+72 } $ with a remainder of $ \color{red}{ -150 } $.