Check if $ x+2 $ is a factor of $ 2x^{2}-3x-2 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&2&-3&-2\\& & -4& \color{black}{14} \\ \hline &\color{blue}{2}&\color{blue}{-7}&\color{orangered}{12} \end{array} $$Because the remainder $ \left( \color{red}{ 12 } \right) $ is not zero, we conclude that the $ x+2 $ is not a factor of $ 2x^{2}-3x-2$.
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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-3&-2\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 2 }&-3&-2\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-3&-2\\& & \color{blue}{-4} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrr}-2&2&\color{orangered}{ -3 }&-2\\& & \color{orangered}{-4} & \\ \hline &2&\color{orangered}{-7}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&2&-3&-2\\& & -4& \color{blue}{14} \\ \hline &2&\color{blue}{-7}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 14 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrr}-2&2&-3&\color{orangered}{ -2 }\\& & -4& \color{orangered}{14} \\ \hline &\color{blue}{2}&\color{blue}{-7}&\color{orangered}{12} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 12 }\right)$.