Check if $ x-5 $ is a factor of $ -x^{3}+8x+20 $.
The synthetic division table is:
$$ \begin{array}{c|rrrr}5&-1&0&8&20\\& & -5& -25& \color{black}{-85} \\ \hline &\color{blue}{-1}&\color{blue}{-5}&\color{blue}{-17}&\color{orangered}{-65} \end{array} $$Because the remainder $ \left( \color{red}{ -65 } \right) $ is not zero, we conclude that the $ x-5 $ is not a factor of $ -x^{3}+8x+20$.
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.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&0&8&20\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}5&\color{orangered}{ -1 }&0&8&20\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&0&8&20\\& & \color{blue}{-5} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}5&-1&\color{orangered}{ 0 }&8&20\\& & \color{orangered}{-5} & & \\ \hline &-1&\color{orangered}{-5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&0&8&20\\& & -5& \color{blue}{-25} & \\ \hline &-1&\color{blue}{-5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrr}5&-1&0&\color{orangered}{ 8 }&20\\& & -5& \color{orangered}{-25} & \\ \hline &-1&-5&\color{orangered}{-17}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ -85 } $.
$$ \begin{array}{c|rrrr}\color{blue}{5}&-1&0&8&20\\& & -5& -25& \color{blue}{-85} \\ \hline &-1&-5&\color{blue}{-17}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -85 \right) } = \color{orangered}{ -65 } $
$$ \begin{array}{c|rrrr}5&-1&0&8&\color{orangered}{ 20 }\\& & -5& -25& \color{orangered}{-85} \\ \hline &\color{blue}{-1}&\color{blue}{-5}&\color{blue}{-17}&\color{orangered}{-65} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -65 }\right)$.