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