The quadratic trinomial in the numerator can be factored into (x -4)(x - 8).
The quadratic trinomial in the denominator can be factored into (x - 4)(x + 3).
The rational function can be reduced to
f(x) = (x - 8)/(x + 3), x =/= 4
Since the binomials x - 4 from the numerator and denominator cancel out, there is an excluded value in the domain of 4, but there is no asymptote there. The binomial x + 3 in the denominator causes the asymptote.