Check if $ -2 $ is a factor of $ 21x^{2}-27x-18 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}0&21&-27&-18\\& & 0& \color{black}{0} \\ \hline &\color{blue}{21}&\color{blue}{-27}&\color{orangered}{-18} \end{array} $$Because the remainder $ \left( \color{red}{ -18 } \right) $ is not zero, we conclude that the $ x $ is not a factor of $ 21x^{2}-27x-18$.
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 at the left.
$$ \begin{array}{c|rrr}\color{blue}{0}&21&-27&-18\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}0&\color{orangered}{ 21 }&-27&-18\\& & & \\ \hline &\color{orangered}{21}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 21 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&21&-27&-18\\& & \color{blue}{0} & \\ \hline &\color{blue}{21}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ 0 } = \color{orangered}{ -27 } $
$$ \begin{array}{c|rrr}0&21&\color{orangered}{ -27 }&-18\\& & \color{orangered}{0} & \\ \hline &21&\color{orangered}{-27}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -27 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&21&-27&-18\\& & 0& \color{blue}{0} \\ \hline &21&\color{blue}{-27}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 0 } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrr}0&21&-27&\color{orangered}{ -18 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{21}&\color{blue}{-27}&\color{orangered}{-18} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -18 }\right)$.