Divide using synthetic division $$ ( 18x^{3}+24x^{2}+10x+18 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&18&24&10&18\\& & -36& 24& \color{black}{-68} \\ \hline &\color{blue}{18}&\color{blue}{-12}&\color{blue}{34}&\color{orangered}{-50} \end{array} $$The solution is:
$$ \frac{ 18x^{3}+24x^{2}+10x+18 }{ x+2 } = \color{blue}{18x^{2}-12x+34} \color{red}{~-~} \frac{ \color{red}{ 50 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&18&24&10&18\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 18 }&24&10&18\\& & & & \\ \hline &\color{orangered}{18}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 18 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&18&24&10&18\\& & \color{blue}{-36} & & \\ \hline &\color{blue}{18}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrr}-2&18&\color{orangered}{ 24 }&10&18\\& & \color{orangered}{-36} & & \\ \hline &18&\color{orangered}{-12}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&18&24&10&18\\& & -36& \color{blue}{24} & \\ \hline &18&\color{blue}{-12}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 24 } = \color{orangered}{ 34 } $
$$ \begin{array}{c|rrrr}-2&18&24&\color{orangered}{ 10 }&18\\& & -36& \color{orangered}{24} & \\ \hline &18&-12&\color{orangered}{34}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 34 } = \color{blue}{ -68 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&18&24&10&18\\& & -36& 24& \color{blue}{-68} \\ \hline &18&-12&\color{blue}{34}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -68 \right) } = \color{orangered}{ -50 } $
$$ \begin{array}{c|rrrr}-2&18&24&10&\color{orangered}{ 18 }\\& & -36& 24& \color{orangered}{-68} \\ \hline &\color{blue}{18}&\color{blue}{-12}&\color{blue}{34}&\color{orangered}{-50} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 18x^{2}-12x+34 } $ with a remainder of $ \color{red}{ -50 } $.