Divide using synthetic division $$ ( x^{5}-3x^{4}-7x^{3}+7x^{2}+18x+8 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}2&1&-3&-7&7&18&8\\& & 2& -2& -18& -22& \color{black}{-8} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-9}&\color{blue}{-11}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{5}-3x^{4}-7x^{3}+7x^{2}+18x+8 }{ x-2 } = \color{blue}{x^{4}-x^{3}-9x^{2}-11x-4} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&-3&-7&7&18&8\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}2&\color{orangered}{ 1 }&-3&-7&7&18&8\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&-3&-7&7&18&8\\& & \color{blue}{2} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 2 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrr}2&1&\color{orangered}{ -3 }&-7&7&18&8\\& & \color{orangered}{2} & & & & \\ \hline &1&\color{orangered}{-1}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&-3&-7&7&18&8\\& & 2& \color{blue}{-2} & & & \\ \hline &1&\color{blue}{-1}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrrr}2&1&-3&\color{orangered}{ -7 }&7&18&8\\& & 2& \color{orangered}{-2} & & & \\ \hline &1&-1&\color{orangered}{-9}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&-3&-7&7&18&8\\& & 2& -2& \color{blue}{-18} & & \\ \hline &1&-1&\color{blue}{-9}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrrrr}2&1&-3&-7&\color{orangered}{ 7 }&18&8\\& & 2& -2& \color{orangered}{-18} & & \\ \hline &1&-1&-9&\color{orangered}{-11}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ -22 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&-3&-7&7&18&8\\& & 2& -2& -18& \color{blue}{-22} & \\ \hline &1&-1&-9&\color{blue}{-11}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -22 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}2&1&-3&-7&7&\color{orangered}{ 18 }&8\\& & 2& -2& -18& \color{orangered}{-22} & \\ \hline &1&-1&-9&-11&\color{orangered}{-4}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&1&-3&-7&7&18&8\\& & 2& -2& -18& -22& \color{blue}{-8} \\ \hline &1&-1&-9&-11&\color{blue}{-4}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}2&1&-3&-7&7&18&\color{orangered}{ 8 }\\& & 2& -2& -18& -22& \color{orangered}{-8} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-9}&\color{blue}{-11}&\color{blue}{-4}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-x^{3}-9x^{2}-11x-4 } $ with a remainder of $ \color{red}{ 0 } $.