Check if $ -x+3 $ is a factor of $ 2x^{2}-7x-29 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-3&2&-7&-29\\& & -6& \color{black}{39} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{orangered}{10} \end{array} $$Because the remainder $ \left( \color{red}{ 10 } \right) $ is not zero, we conclude that the $ x+3 $ 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 + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-3}&2&-7&-29\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-3&\color{orangered}{ 2 }&-7&-29\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&2&-7&-29\\& & \color{blue}{-6} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrr}-3&2&\color{orangered}{ -7 }&-29\\& & \color{orangered}{-6} & \\ \hline &2&\color{orangered}{-13}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 39 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&2&-7&-29\\& & -6& \color{blue}{39} \\ \hline &2&\color{blue}{-13}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ 39 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrr}-3&2&-7&\color{orangered}{ -29 }\\& & -6& \color{orangered}{39} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{orangered}{10} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 10 }\right)$.