Check if $ x+5 $ is a factor of $ x^{4}+x^{3}-6x^{2}-x+5 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&1&1&-6&-1&5\\& & -5& 20& -70& \color{black}{355} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{14}&\color{blue}{-71}&\color{orangered}{360} \end{array} $$Because the remainder $ \left( \color{red}{ 360 } \right) $ is not zero, we conclude that the $ x+5 $ is not a factor of $ x^{4}+x^{3}-6x^{2}-x+5$.
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&1&-6&-1&5\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 1 }&1&-6&-1&5\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&1&-6&-1&5\\& & \color{blue}{-5} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}-5&1&\color{orangered}{ 1 }&-6&-1&5\\& & \color{orangered}{-5} & & & \\ \hline &1&\color{orangered}{-4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&1&-6&-1&5\\& & -5& \color{blue}{20} & & \\ \hline &1&\color{blue}{-4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 20 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrrr}-5&1&1&\color{orangered}{ -6 }&-1&5\\& & -5& \color{orangered}{20} & & \\ \hline &1&-4&\color{orangered}{14}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 14 } = \color{blue}{ -70 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&1&-6&-1&5\\& & -5& 20& \color{blue}{-70} & \\ \hline &1&-4&\color{blue}{14}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -70 \right) } = \color{orangered}{ -71 } $
$$ \begin{array}{c|rrrrr}-5&1&1&-6&\color{orangered}{ -1 }&5\\& & -5& 20& \color{orangered}{-70} & \\ \hline &1&-4&14&\color{orangered}{-71}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -71 \right) } = \color{blue}{ 355 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&1&-6&-1&5\\& & -5& 20& -70& \color{blue}{355} \\ \hline &1&-4&14&\color{blue}{-71}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 355 } = \color{orangered}{ 360 } $
$$ \begin{array}{c|rrrrr}-5&1&1&-6&-1&\color{orangered}{ 5 }\\& & -5& 20& -70& \color{orangered}{355} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{14}&\color{blue}{-71}&\color{orangered}{360} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 360 }\right)$.