The synthetic division table is:
$$ \begin{array}{c|rrrrr}5&5&0&2&-15&10\\& & 25& 125& 635& \color{black}{3100} \\ \hline &\color{blue}{5}&\color{blue}{25}&\color{blue}{127}&\color{blue}{620}&\color{orangered}{3110} \end{array} $$The solution is:
$$ \frac{ 5x^{4}+2x^{2}-15x+10 }{ x-5 } = \color{blue}{5x^{3}+25x^{2}+127x+620} ~+~ \frac{ \color{red}{ 3110 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&5&0&2&-15&10\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}5&\color{orangered}{ 5 }&0&2&-15&10\\& & & & & \\ \hline &\color{orangered}{5}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 5 } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&5&0&2&-15&10\\& & \color{blue}{25} & & & \\ \hline &\color{blue}{5}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 25 } = \color{orangered}{ 25 } $
$$ \begin{array}{c|rrrrr}5&5&\color{orangered}{ 0 }&2&-15&10\\& & \color{orangered}{25} & & & \\ \hline &5&\color{orangered}{25}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 25 } = \color{blue}{ 125 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&5&0&2&-15&10\\& & 25& \color{blue}{125} & & \\ \hline &5&\color{blue}{25}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 125 } = \color{orangered}{ 127 } $
$$ \begin{array}{c|rrrrr}5&5&0&\color{orangered}{ 2 }&-15&10\\& & 25& \color{orangered}{125} & & \\ \hline &5&25&\color{orangered}{127}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 127 } = \color{blue}{ 635 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&5&0&2&-15&10\\& & 25& 125& \color{blue}{635} & \\ \hline &5&25&\color{blue}{127}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 635 } = \color{orangered}{ 620 } $
$$ \begin{array}{c|rrrrr}5&5&0&2&\color{orangered}{ -15 }&10\\& & 25& 125& \color{orangered}{635} & \\ \hline &5&25&127&\color{orangered}{620}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 620 } = \color{blue}{ 3100 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&5&0&2&-15&10\\& & 25& 125& 635& \color{blue}{3100} \\ \hline &5&25&127&\color{blue}{620}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 3100 } = \color{orangered}{ 3110 } $
$$ \begin{array}{c|rrrrr}5&5&0&2&-15&\color{orangered}{ 10 }\\& & 25& 125& 635& \color{orangered}{3100} \\ \hline &\color{blue}{5}&\color{blue}{25}&\color{blue}{127}&\color{blue}{620}&\color{orangered}{3110} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{3}+25x^{2}+127x+620 } $ with a remainder of $ \color{red}{ 3110 } $.