Divide using synthetic division $$ ( 18x^{3}+20x^{2}+10x+8 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&18&20&10&8\\& & -36& 32& \color{black}{-84} \\ \hline &\color{blue}{18}&\color{blue}{-16}&\color{blue}{42}&\color{orangered}{-76} \end{array} $$The solution is:
$$ \frac{ 18x^{3}+20x^{2}+10x+8 }{ x+2 } = \color{blue}{18x^{2}-16x+42} \color{red}{~-~} \frac{ \color{red}{ 76 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&18&20&10&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 18 }&20&10&8\\& & & & \\ \hline &\color{orangered}{18}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 18 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&18&20&10&8\\& & \color{blue}{-36} & & \\ \hline &\color{blue}{18}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrr}-2&18&\color{orangered}{ 20 }&10&8\\& & \color{orangered}{-36} & & \\ \hline &18&\color{orangered}{-16}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&18&20&10&8\\& & -36& \color{blue}{32} & \\ \hline &18&\color{blue}{-16}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 32 } = \color{orangered}{ 42 } $
$$ \begin{array}{c|rrrr}-2&18&20&\color{orangered}{ 10 }&8\\& & -36& \color{orangered}{32} & \\ \hline &18&-16&\color{orangered}{42}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 42 } = \color{blue}{ -84 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&18&20&10&8\\& & -36& 32& \color{blue}{-84} \\ \hline &18&-16&\color{blue}{42}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -84 \right) } = \color{orangered}{ -76 } $
$$ \begin{array}{c|rrrr}-2&18&20&10&\color{orangered}{ 8 }\\& & -36& 32& \color{orangered}{-84} \\ \hline &\color{blue}{18}&\color{blue}{-16}&\color{blue}{42}&\color{orangered}{-76} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 18x^{2}-16x+42 } $ with a remainder of $ \color{red}{ -76 } $.