Check if $ x-2 $ is a factor of $ x^{2}-19x-25 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&1&-19&-25\\& & 2& \color{black}{-34} \\ \hline &\color{blue}{1}&\color{blue}{-17}&\color{orangered}{-59} \end{array} $$Because the remainder $ \left( \color{red}{ -59 } \right) $ is not zero, we conclude that the $ x-2 $ is not a factor of $ x^{2}-19x-25$.
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 -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{2}&1&-19&-25\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}2&\color{orangered}{ 1 }&-19&-25\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&1&-19&-25\\& & \color{blue}{2} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 2 } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrr}2&1&\color{orangered}{ -19 }&-25\\& & \color{orangered}{2} & \\ \hline &1&\color{orangered}{-17}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ -34 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&1&-19&-25\\& & 2& \color{blue}{-34} \\ \hline &1&\color{blue}{-17}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ \left( -34 \right) } = \color{orangered}{ -59 } $
$$ \begin{array}{c|rrr}2&1&-19&\color{orangered}{ -25 }\\& & 2& \color{orangered}{-34} \\ \hline &\color{blue}{1}&\color{blue}{-17}&\color{orangered}{-59} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -59 }\right)$.