Divide using synthetic division $$ ( 4x^{5}-10x^{4}+8x^{3}+x^{2}-8 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}2&4&-10&8&1&0&-8\\& & 8& -4& 8& 18& \color{black}{36} \\ \hline &\color{blue}{4}&\color{blue}{-2}&\color{blue}{4}&\color{blue}{9}&\color{blue}{18}&\color{orangered}{28} \end{array} $$The solution is:
$$ \frac{ 4x^{5}-10x^{4}+8x^{3}+x^{2}-8 }{ x-2 } = \color{blue}{4x^{4}-2x^{3}+4x^{2}+9x+18} ~+~ \frac{ \color{red}{ 28 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&-10&8&1&0&-8\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}2&\color{orangered}{ 4 }&-10&8&1&0&-8\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&-10&8&1&0&-8\\& & \color{blue}{8} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 8 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrr}2&4&\color{orangered}{ -10 }&8&1&0&-8\\& & \color{orangered}{8} & & & & \\ \hline &4&\color{orangered}{-2}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&-10&8&1&0&-8\\& & 8& \color{blue}{-4} & & & \\ \hline &4&\color{blue}{-2}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}2&4&-10&\color{orangered}{ 8 }&1&0&-8\\& & 8& \color{orangered}{-4} & & & \\ \hline &4&-2&\color{orangered}{4}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 4 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&-10&8&1&0&-8\\& & 8& -4& \color{blue}{8} & & \\ \hline &4&-2&\color{blue}{4}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 8 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrrr}2&4&-10&8&\color{orangered}{ 1 }&0&-8\\& & 8& -4& \color{orangered}{8} & & \\ \hline &4&-2&4&\color{orangered}{9}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 9 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&-10&8&1&0&-8\\& & 8& -4& 8& \color{blue}{18} & \\ \hline &4&-2&4&\color{blue}{9}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrr}2&4&-10&8&1&\color{orangered}{ 0 }&-8\\& & 8& -4& 8& \color{orangered}{18} & \\ \hline &4&-2&4&9&\color{orangered}{18}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 18 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&4&-10&8&1&0&-8\\& & 8& -4& 8& 18& \color{blue}{36} \\ \hline &4&-2&4&9&\color{blue}{18}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 36 } = \color{orangered}{ 28 } $
$$ \begin{array}{c|rrrrrr}2&4&-10&8&1&0&\color{orangered}{ -8 }\\& & 8& -4& 8& 18& \color{orangered}{36} \\ \hline &\color{blue}{4}&\color{blue}{-2}&\color{blue}{4}&\color{blue}{9}&\color{blue}{18}&\color{orangered}{28} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{4}-2x^{3}+4x^{2}+9x+18 } $ with a remainder of $ \color{red}{ 28 } $.