Check if $ -1 $ is a factor of $ 4x^{5}-8x^{4}-5x^{3}+10x^{2}+x-2 $.
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}0&4&-8&-5&10&1&-2\\& & 0& 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{4}&\color{blue}{-8}&\color{blue}{-5}&\color{blue}{10}&\color{blue}{1}&\color{orangered}{-2} \end{array} $$Because the remainder $ \left( \color{red}{ -2 } \right) $ is not zero, we conclude that the $ x $ is not a factor of $ 4x^{5}-8x^{4}-5x^{3}+10x^{2}+x-2$.
First we need to create a synthetic division table.
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-8&-5&10&1&-2\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}0&\color{orangered}{ 4 }&-8&-5&10&1&-2\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 4 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-8&-5&10&1&-2\\& & \color{blue}{0} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 0 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrrr}0&4&\color{orangered}{ -8 }&-5&10&1&-2\\& & \color{orangered}{0} & & & & \\ \hline &4&\color{orangered}{-8}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-8&-5&10&1&-2\\& & 0& \color{blue}{0} & & & \\ \hline &4&\color{blue}{-8}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 0 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrr}0&4&-8&\color{orangered}{ -5 }&10&1&-2\\& & 0& \color{orangered}{0} & & & \\ \hline &4&-8&\color{orangered}{-5}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-8&-5&10&1&-2\\& & 0& 0& \color{blue}{0} & & \\ \hline &4&-8&\color{blue}{-5}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 0 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrrr}0&4&-8&-5&\color{orangered}{ 10 }&1&-2\\& & 0& 0& \color{orangered}{0} & & \\ \hline &4&-8&-5&\color{orangered}{10}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 10 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-8&-5&10&1&-2\\& & 0& 0& 0& \color{blue}{0} & \\ \hline &4&-8&-5&\color{blue}{10}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 0 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrr}0&4&-8&-5&10&\color{orangered}{ 1 }&-2\\& & 0& 0& 0& \color{orangered}{0} & \\ \hline &4&-8&-5&10&\color{orangered}{1}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&4&-8&-5&10&1&-2\\& & 0& 0& 0& 0& \color{blue}{0} \\ \hline &4&-8&-5&10&\color{blue}{1}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 0 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrr}0&4&-8&-5&10&1&\color{orangered}{ -2 }\\& & 0& 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{4}&\color{blue}{-8}&\color{blue}{-5}&\color{blue}{10}&\color{blue}{1}&\color{orangered}{-2} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -2 }\right)$.