1. **Problem:** Find the asymptotes of the rational function $$y = \frac{x^2 + 12x + 35}{3x^2 + 19x + 20}$$. 2. **Vertical asymptotes:** These occur where the denominator is zero (and numerator is not zero at those points). 3. Factor the denominator: $$3x^2 + 19x + 20 = (3x + 4)(x + 5)$$ Set each factor to zero: $$3x + 4 = 0 \Rightarrow x = -\frac{4}{3}$$ $$x + 5 = 0 \Rightarrow x = -5$$ 4. Check numerator at these points: $$x^2 + 12x + 35 = (x + 7)(x + 5)$$ At $$x = -5$$ numerator is zero, so this is a removable discontinuity, not a vertical asymptote. At $$x = -\frac{4}{3}$$ numerator is not zero, so vertical asymptote at $$x = -\frac{4}{3}$$. 5. **Horizontal asymptote:** Compare degrees of numerator and denominator. Both numerator and denominator are degree 2. 6. Horizontal asymptote is ratio of leading coefficients: $$y = \frac{1}{3}$$ **Final answer:** - Vertical asymptote at $$x = -\frac{4}{3}$$ - Horizontal asymptote at $$y = \frac{1}{3}$$