Check if $ -2x+1 $ is a factor of $ 2x^{2}-7x-29 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-1&2&-7&-29\\& & -2& \color{black}{9} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{orangered}{-20} \end{array} $$Because the remainder $ \left( \color{red}{ -20 } \right) $ is not zero, we conclude that the $ x+1 $ is not a factor of $ 2x^{2}-7x-29$.
Explanation
First we need to create a synthetic division table.
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.
$$ \begin{array}{c|rrr}\color{blue}{-1}&2&-7&-29\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-1&\color{orangered}{ 2 }&-7&-29\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 2 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&2&-7&-29\\& & \color{blue}{-2} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrr}-1&2&\color{orangered}{ -7 }&-29\\& & \color{orangered}{-2} & \\ \hline &2&\color{orangered}{-9}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&2&-7&-29\\& & -2& \color{blue}{9} \\ \hline &2&\color{blue}{-9}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ 9 } = \color{orangered}{ -20 } $
$$ \begin{array}{c|rrr}-1&2&-7&\color{orangered}{ -29 }\\& & -2& \color{orangered}{9} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{orangered}{-20} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -20 }\right)$.