Divide using synthetic division $$ ( x^{3}-6x^{2}+11x-6 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&1&-6&11&-6\\& & -1& 7& \color{black}{-18} \\ \hline &\color{blue}{1}&\color{blue}{-7}&\color{blue}{18}&\color{orangered}{-24} \end{array} $$The solution is:
$$ \frac{ x^{3}-6x^{2}+11x-6 }{ x+1 } = \color{blue}{x^{2}-7x+18} \color{red}{~-~} \frac{ \color{red}{ 24 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&1&-6&11&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 1 }&-6&11&-6\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&1&-6&11&-6\\& & \color{blue}{-1} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}-1&1&\color{orangered}{ -6 }&11&-6\\& & \color{orangered}{-1} & & \\ \hline &1&\color{orangered}{-7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 7 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&1&-6&11&-6\\& & -1& \color{blue}{7} & \\ \hline &1&\color{blue}{-7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ 7 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}-1&1&-6&\color{orangered}{ 11 }&-6\\& & -1& \color{orangered}{7} & \\ \hline &1&-7&\color{orangered}{18}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 18 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&1&-6&11&-6\\& & -1& 7& \color{blue}{-18} \\ \hline &1&-7&\color{blue}{18}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrr}-1&1&-6&11&\color{orangered}{ -6 }\\& & -1& 7& \color{orangered}{-18} \\ \hline &\color{blue}{1}&\color{blue}{-7}&\color{blue}{18}&\color{orangered}{-24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-7x+18 } $ with a remainder of $ \color{red}{ -24 } $.