Divide using synthetic division $$ ( 2x^{4}+13x^{3}+21x^{2}-3x-18 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&2&13&21&-3&-18\\& & -4& -18& -6& \color{black}{18} \\ \hline &\color{blue}{2}&\color{blue}{9}&\color{blue}{3}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{4}+13x^{3}+21x^{2}-3x-18 }{ x+2 } = \color{blue}{2x^{3}+9x^{2}+3x-9} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&13&21&-3&-18\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 2 }&13&21&-3&-18\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&13&21&-3&-18\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}-2&2&\color{orangered}{ 13 }&21&-3&-18\\& & \color{orangered}{-4} & & & \\ \hline &2&\color{orangered}{9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 9 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&13&21&-3&-18\\& & -4& \color{blue}{-18} & & \\ \hline &2&\color{blue}{9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}-2&2&13&\color{orangered}{ 21 }&-3&-18\\& & -4& \color{orangered}{-18} & & \\ \hline &2&9&\color{orangered}{3}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&13&21&-3&-18\\& & -4& -18& \color{blue}{-6} & \\ \hline &2&9&\color{blue}{3}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-2&2&13&21&\color{orangered}{ -3 }&-18\\& & -4& -18& \color{orangered}{-6} & \\ \hline &2&9&3&\color{orangered}{-9}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&13&21&-3&-18\\& & -4& -18& -6& \color{blue}{18} \\ \hline &2&9&3&\color{blue}{-9}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 18 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-2&2&13&21&-3&\color{orangered}{ -18 }\\& & -4& -18& -6& \color{orangered}{18} \\ \hline &\color{blue}{2}&\color{blue}{9}&\color{blue}{3}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+9x^{2}+3x-9 } $ with a remainder of $ \color{red}{ 0 } $.