Answer
Based on the synthetic division shown below, the equation of the slant asymptote of
$f\left( x \right)=\frac{3{{x}^{2}}-7x+5}{x-4}$ is $y=3x+5$.
Work Step by Step
Divide $3{{x}^{2}}-7x+5$ by $x-4$.
Use synthetic division,
$\begin{align}
& \left. {\underline {\,
4 \,}}\! \right| \ 3\ -7\ 5 \\
& \ \ \underline{\ \ \ \ 12\ 20} \\
& \ \ 3\ \ \ 5\ \ 25 \\
\end{align}$
The reminder is 25.
So, $f\left( x \right)=3x+5+\frac{25}{x-4}.$
The equation of the asymptote is $y=3x+5$.
