The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&5&-24&17&-5&4\\& & 20& -16& 4& \color{black}{-4} \\ \hline &\color{blue}{5}&\color{blue}{-4}&\color{blue}{1}&\color{blue}{-1}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 5x^{4}-24x^{3}+17x^{2}-5x+4 }{ x-4 } = \color{blue}{5x^{3}-4x^{2}+x-1} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&5&-24&17&-5&4\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 5 }&-24&17&-5&4\\& & & & & \\ \hline &\color{orangered}{5}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 5 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&5&-24&17&-5&4\\& & \color{blue}{20} & & & \\ \hline &\color{blue}{5}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 20 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}4&5&\color{orangered}{ -24 }&17&-5&4\\& & \color{orangered}{20} & & & \\ \hline &5&\color{orangered}{-4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&5&-24&17&-5&4\\& & 20& \color{blue}{-16} & & \\ \hline &5&\color{blue}{-4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrr}4&5&-24&\color{orangered}{ 17 }&-5&4\\& & 20& \color{orangered}{-16} & & \\ \hline &5&-4&\color{orangered}{1}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&5&-24&17&-5&4\\& & 20& -16& \color{blue}{4} & \\ \hline &5&-4&\color{blue}{1}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 4 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}4&5&-24&17&\color{orangered}{ -5 }&4\\& & 20& -16& \color{orangered}{4} & \\ \hline &5&-4&1&\color{orangered}{-1}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&5&-24&17&-5&4\\& & 20& -16& 4& \color{blue}{-4} \\ \hline &5&-4&1&\color{blue}{-1}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}4&5&-24&17&-5&\color{orangered}{ 4 }\\& & 20& -16& 4& \color{orangered}{-4} \\ \hline &\color{blue}{5}&\color{blue}{-4}&\color{blue}{1}&\color{blue}{-1}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{3}-4x^{2}+x-1 } $ with a remainder of $ \color{red}{ 0 } $.