Check if $ x+5 $ is a factor of $ -4x^{2}+3x+7 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&-4&3&7\\& & 20& \color{black}{-115} \\ \hline &\color{blue}{-4}&\color{blue}{23}&\color{orangered}{-108} \end{array} $$Because the remainder $ \left( \color{red}{ -108 } \right) $ is not zero, we conclude that the $ x+5 $ is not a factor of $ -4x^{2}+3x+7$.
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 + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-5}&-4&3&7\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ -4 }&3&7\\& & & \\ \hline &\color{orangered}{-4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&-4&3&7\\& & \color{blue}{20} & \\ \hline &\color{blue}{-4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 20 } = \color{orangered}{ 23 } $
$$ \begin{array}{c|rrr}-5&-4&\color{orangered}{ 3 }&7\\& & \color{orangered}{20} & \\ \hline &-4&\color{orangered}{23}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 23 } = \color{blue}{ -115 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&-4&3&7\\& & 20& \color{blue}{-115} \\ \hline &-4&\color{blue}{23}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -115 \right) } = \color{orangered}{ -108 } $
$$ \begin{array}{c|rrr}-5&-4&3&\color{orangered}{ 7 }\\& & 20& \color{orangered}{-115} \\ \hline &\color{blue}{-4}&\color{blue}{23}&\color{orangered}{-108} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -108 }\right)$.