The asymptotes of q(x) = (5x^2 - 13x - 6)/(2x^2 - 5x - 3) have to be determined.
The vertical asymptotes are given by x = a where a is the zeros of the denominator.
2x^2 - 5x - 3 = 0
=> 2x^2 - 6x + x - 3 = 0
=> 2x(x - 3) + 1(x - 3) = 0
=> (2x + 1)(x - 3) = 0
=> x = -1/2 and x = 3
The horizontal asymptote is given by y = (leading coefficient of numerator)/(leading coefficient of denominator). This gives y = 5/2
As the numerator and denominator have the same degree there is no slant asymptote.