Divide using synthetic division $$ ( 9x^{4}-20x^{2}+23x-18 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&9&0&-20&23&-18\\& & -18& 36& -32& \color{black}{18} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{blue}{16}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 9x^{4}-20x^{2}+23x-18 }{ x+2 } = \color{blue}{9x^{3}-18x^{2}+16x-9} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&0&-20&23&-18\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 9 }&0&-20&23&-18\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&0&-20&23&-18\\& & \color{blue}{-18} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrr}-2&9&\color{orangered}{ 0 }&-20&23&-18\\& & \color{orangered}{-18} & & & \\ \hline &9&\color{orangered}{-18}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&0&-20&23&-18\\& & -18& \color{blue}{36} & & \\ \hline &9&\color{blue}{-18}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 36 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrr}-2&9&0&\color{orangered}{ -20 }&23&-18\\& & -18& \color{orangered}{36} & & \\ \hline &9&-18&\color{orangered}{16}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 16 } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&0&-20&23&-18\\& & -18& 36& \color{blue}{-32} & \\ \hline &9&-18&\color{blue}{16}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-2&9&0&-20&\color{orangered}{ 23 }&-18\\& & -18& 36& \color{orangered}{-32} & \\ \hline &9&-18&16&\color{orangered}{-9}& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&0&-20&23&-18\\& & -18& 36& -32& \color{blue}{18} \\ \hline &9&-18&16&\color{blue}{-9}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 18 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-2&9&0&-20&23&\color{orangered}{ -18 }\\& & -18& 36& -32& \color{orangered}{18} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{blue}{16}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}-18x^{2}+16x-9 } $ with a remainder of $ \color{red}{ 0 } $.