Divide using synthetic division $$ ( 18x^{3}+10x^{2}+27x+18 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&18&10&27&18\\& & -108& 588& \color{black}{-3690} \\ \hline &\color{blue}{18}&\color{blue}{-98}&\color{blue}{615}&\color{orangered}{-3672} \end{array} $$The solution is:
$$ \frac{ 18x^{3}+10x^{2}+27x+18 }{ x+6 } = \color{blue}{18x^{2}-98x+615} \color{red}{~-~} \frac{ \color{red}{ 3672 } }{ x+6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&18&10&27&18\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 18 }&10&27&18\\& & & & \\ \hline &\color{orangered}{18}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 18 } = \color{blue}{ -108 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&18&10&27&18\\& & \color{blue}{-108} & & \\ \hline &\color{blue}{18}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -108 \right) } = \color{orangered}{ -98 } $
$$ \begin{array}{c|rrrr}-6&18&\color{orangered}{ 10 }&27&18\\& & \color{orangered}{-108} & & \\ \hline &18&\color{orangered}{-98}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -98 \right) } = \color{blue}{ 588 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&18&10&27&18\\& & -108& \color{blue}{588} & \\ \hline &18&\color{blue}{-98}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 27 } + \color{orangered}{ 588 } = \color{orangered}{ 615 } $
$$ \begin{array}{c|rrrr}-6&18&10&\color{orangered}{ 27 }&18\\& & -108& \color{orangered}{588} & \\ \hline &18&-98&\color{orangered}{615}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 615 } = \color{blue}{ -3690 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&18&10&27&18\\& & -108& 588& \color{blue}{-3690} \\ \hline &18&-98&\color{blue}{615}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -3690 \right) } = \color{orangered}{ -3672 } $
$$ \begin{array}{c|rrrr}-6&18&10&27&\color{orangered}{ 18 }\\& & -108& 588& \color{orangered}{-3690} \\ \hline &\color{blue}{18}&\color{blue}{-98}&\color{blue}{615}&\color{orangered}{-3672} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 18x^{2}-98x+615 } $ with a remainder of $ \color{red}{ -3672 } $.