Answer
vertical asymptote $x=-\frac{1}{3}$,
horizontal asymptote $y=\frac{2}{3}$,
oblique asymptote $none$.
Work Step by Step
Step 1. Factor the function $Q(x)=\frac{2x^2-5x-12}{3x^2-11x-4}=\frac{(x-4)(2x+3)}{(x-4)(3x+1)}=\frac{2x+3}{3x+1}, (x\ne4)$,
Step 2. we can find the vertical asymptote $x=-\frac{1}{3}$,
Step 3. we can find the horizontal asymptote $y=\frac{2}{3}$,
Step 4. we can find the oblique asymptote $none$.