Divide using synthetic division $$ ( x^{3}-10x^{2}+7x+18 ) \div ( x+1 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&1&-10&7&18\\& & -1& 11& \color{black}{-18} \\ \hline &\color{blue}{1}&\color{blue}{-11}&\color{blue}{18}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ x^{3}-10x^{2}+7x+18 }{ x+1 } = \color{blue}{x^{2}-11x+18} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&1&-10&7&18\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 1 }&-10&7&18\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&1&-10&7&18\\& & \color{blue}{-1} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrr}-1&1&\color{orangered}{ -10 }&7&18\\& & \color{orangered}{-1} & & \\ \hline &1&\color{orangered}{-11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 11 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&1&-10&7&18\\& & -1& \color{blue}{11} & \\ \hline &1&\color{blue}{-11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 11 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}-1&1&-10&\color{orangered}{ 7 }&18\\& & -1& \color{orangered}{11} & \\ \hline &1&-11&\color{orangered}{18}& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&1&-10&7&18\\& & -1& 11& \color{blue}{-18} \\ \hline &1&-11&\color{blue}{18}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-1&1&-10&7&\color{orangered}{ 18 }\\& & -1& 11& \color{orangered}{-18} \\ \hline &\color{blue}{1}&\color{blue}{-11}&\color{blue}{18}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-11x+18 } $ with a remainder of $ \color{red}{ 0 } $.