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