Check if $ x-4 $ is a factor of $ -2x^{2}+27x-94 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}4&-2&27&-94\\& & -8& \color{black}{76} \\ \hline &\color{blue}{-2}&\color{blue}{19}&\color{orangered}{-18} \end{array} $$Because the remainder $ \left( \color{red}{ -18 } \right) $ is not zero, we conclude that the $ x-4 $ is not a factor of $ -2x^{2}+27x-94$.
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 -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{4}&-2&27&-94\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ -2 }&27&-94\\& & & \\ \hline &\color{orangered}{-2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&-2&27&-94\\& & \color{blue}{-8} & \\ \hline &\color{blue}{-2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 27 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrr}4&-2&\color{orangered}{ 27 }&-94\\& & \color{orangered}{-8} & \\ \hline &-2&\color{orangered}{19}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 19 } = \color{blue}{ 76 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&-2&27&-94\\& & -8& \color{blue}{76} \\ \hline &-2&\color{blue}{19}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -94 } + \color{orangered}{ 76 } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrr}4&-2&27&\color{orangered}{ -94 }\\& & -8& \color{orangered}{76} \\ \hline &\color{blue}{-2}&\color{blue}{19}&\color{orangered}{-18} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -18 }\right)$.