Divide using synthetic division $$ ( 5x^{3}-3x+8 ) \div ( x-4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}4&5&0&-3&8\\& & 20& 80& \color{black}{308} \\ \hline &\color{blue}{5}&\color{blue}{20}&\color{blue}{77}&\color{orangered}{316} \end{array} $$The solution is:
$$ \dfrac{ 5x^{3}-3x+8 }{ x-4 } = \color{blue}{5x^{2}+20x+77} ~+~ \dfrac{ \color{red}{ 316 } }{ x-4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{4}&5&0&-3&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}4&\color{orangered}{ 5 }&0&-3&8\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&5&0&-3&8\\& & \color{blue}{20} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 20 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrrr}4&5&\color{orangered}{ 0 }&-3&8\\& & \color{orangered}{20} & & \\ \hline &5&\color{orangered}{20}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 20 } = \color{blue}{ 80 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&5&0&-3&8\\& & 20& \color{blue}{80} & \\ \hline &5&\color{blue}{20}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 80 } = \color{orangered}{ 77 } $
$$ \begin{array}{c|rrrr}4&5&0&\color{orangered}{ -3 }&8\\& & 20& \color{orangered}{80} & \\ \hline &5&20&\color{orangered}{77}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 77 } = \color{blue}{ 308 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&5&0&-3&8\\& & 20& 80& \color{blue}{308} \\ \hline &5&20&\color{blue}{77}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 308 } = \color{orangered}{ 316 } $
$$ \begin{array}{c|rrrr}4&5&0&-3&\color{orangered}{ 8 }\\& & 20& 80& \color{orangered}{308} \\ \hline &\color{blue}{5}&\color{blue}{20}&\color{blue}{77}&\color{orangered}{316} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}+20x+77 } $ with a remainder of $ \color{red}{ 316 } $.