Divide using synthetic division $$ ( 5x^{2}-17x-15 ) \div ( x-4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}4&5&-17&-15\\& & 20& \color{black}{12} \\ \hline &\color{blue}{5}&\color{blue}{3}&\color{orangered}{-3} \end{array} $$The solution is:
$$ \dfrac{ 5x^{2}-17x-15 }{ x-4 } = \color{blue}{5x+3} \color{red}{~-~} \dfrac{ \color{red}{ 3 } }{ x-4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{4}&5&-17&-15\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}4&\color{orangered}{ 5 }&-17&-15\\& & & \\ \hline &\color{orangered}{5}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&5&-17&-15\\& & \color{blue}{20} & \\ \hline &\color{blue}{5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 20 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrr}4&5&\color{orangered}{ -17 }&-15\\& & \color{orangered}{20} & \\ \hline &5&\color{orangered}{3}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 3 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrr}\color{blue}{4}&5&-17&-15\\& & 20& \color{blue}{12} \\ \hline &5&\color{blue}{3}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 12 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrr}4&5&-17&\color{orangered}{ -15 }\\& & 20& \color{orangered}{12} \\ \hline &\color{blue}{5}&\color{blue}{3}&\color{orangered}{-3} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x+3 } $ with a remainder of $ \color{red}{ -3 } $.