Divide using synthetic division $$ ( 4x^{3}-x^{2}+9 ) \div ( x+2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&4&-1&0&9\\& & -8& 18& \color{black}{-36} \\ \hline &\color{blue}{4}&\color{blue}{-9}&\color{blue}{18}&\color{orangered}{-27} \end{array} $$The solution is:
$$ \dfrac{ 4x^{3}-x^{2}+9 }{ x+2 } = \color{blue}{4x^{2}-9x+18} \color{red}{~-~} \dfrac{ \color{red}{ 27 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&4&-1&0&9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 4 }&-1&0&9\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&4&-1&0&9\\& & \color{blue}{-8} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}-2&4&\color{orangered}{ -1 }&0&9\\& & \color{orangered}{-8} & & \\ \hline &4&\color{orangered}{-9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&4&-1&0&9\\& & -8& \color{blue}{18} & \\ \hline &4&\color{blue}{-9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}-2&4&-1&\color{orangered}{ 0 }&9\\& & -8& \color{orangered}{18} & \\ \hline &4&-9&\color{orangered}{18}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 18 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&4&-1&0&9\\& & -8& 18& \color{blue}{-36} \\ \hline &4&-9&\color{blue}{18}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -27 } $
$$ \begin{array}{c|rrrr}-2&4&-1&0&\color{orangered}{ 9 }\\& & -8& 18& \color{orangered}{-36} \\ \hline &\color{blue}{4}&\color{blue}{-9}&\color{blue}{18}&\color{orangered}{-27} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}-9x+18 } $ with a remainder of $ \color{red}{ -27 } $.