Divide using synthetic division $$ ( x^{4}-x^{3}-11x^{2}+9x+18 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-1&1&-1&-11&9&18\\& & -1& 2& 9& \color{black}{-18} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{-9}&\color{blue}{18}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}-x^{3}-11x^{2}+9x+18 }{ x+1 } = \color{blue}{x^{3}-2x^{2}-9x+18} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&1&-1&-11&9&18\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ 1 }&-1&-11&9&18\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&1&-1&-11&9&18\\& & \color{blue}{-1} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-1&1&\color{orangered}{ -1 }&-11&9&18\\& & \color{orangered}{-1} & & & \\ \hline &1&\color{orangered}{-2}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&1&-1&-11&9&18\\& & -1& \color{blue}{2} & & \\ \hline &1&\color{blue}{-2}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 2 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-1&1&-1&\color{orangered}{ -11 }&9&18\\& & -1& \color{orangered}{2} & & \\ \hline &1&-2&\color{orangered}{-9}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&1&-1&-11&9&18\\& & -1& 2& \color{blue}{9} & \\ \hline &1&-2&\color{blue}{-9}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 9 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrr}-1&1&-1&-11&\color{orangered}{ 9 }&18\\& & -1& 2& \color{orangered}{9} & \\ \hline &1&-2&-9&\color{orangered}{18}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 18 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&1&-1&-11&9&18\\& & -1& 2& 9& \color{blue}{-18} \\ \hline &1&-2&-9&\color{blue}{18}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-1&1&-1&-11&9&\color{orangered}{ 18 }\\& & -1& 2& 9& \color{orangered}{-18} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{-9}&\color{blue}{18}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-2x^{2}-9x+18 } $ with a remainder of $ \color{red}{ 0 } $.