Divide using synthetic division $$ ( x^{4}+2x^{3}-12x^{2}-18x+27 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&1&2&-12&-18&27\\& & -3& 3& 27& \color{black}{-27} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-9}&\color{blue}{9}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}+2x^{3}-12x^{2}-18x+27 }{ x+3 } = \color{blue}{x^{3}-x^{2}-9x+9} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&2&-12&-18&27\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&2&-12&-18&27\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&2&-12&-18&27\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ 2 }&-12&-18&27\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{-1}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&2&-12&-18&27\\& & -3& \color{blue}{3} & & \\ \hline &1&\color{blue}{-1}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 3 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-3&1&2&\color{orangered}{ -12 }&-18&27\\& & -3& \color{orangered}{3} & & \\ \hline &1&-1&\color{orangered}{-9}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&2&-12&-18&27\\& & -3& 3& \color{blue}{27} & \\ \hline &1&-1&\color{blue}{-9}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 27 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}-3&1&2&-12&\color{orangered}{ -18 }&27\\& & -3& 3& \color{orangered}{27} & \\ \hline &1&-1&-9&\color{orangered}{9}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 9 } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&2&-12&-18&27\\& & -3& 3& 27& \color{blue}{-27} \\ \hline &1&-1&-9&\color{blue}{9}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 27 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-3&1&2&-12&-18&\color{orangered}{ 27 }\\& & -3& 3& 27& \color{orangered}{-27} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-9}&\color{blue}{9}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-x^{2}-9x+9 } $ with a remainder of $ \color{red}{ 0 } $.