1. **State the problem:** We are given the function $$F(x) = a + \frac{b}{x+1} + \frac{c}{x-3}$$ and we want to understand its structure and behavior. 2. **Formula and rules:** This function is a rational function composed of a constant term $$a$$ and two rational terms with denominators $$x+1$$ and $$x-3$$. 3. **Important points:** The function is undefined where the denominators are zero, i.e., at $$x = -1$$ and $$x = 3$$. These are vertical asymptotes. 4. **Combine terms:** To analyze or simplify, we can write the function over a common denominator: $$F(x) = a + \frac{b}{x+1} + \frac{c}{x-3} = a + \frac{b(x-3)}{(x+1)(x-3)} + \frac{c(x+1)}{(x-3)(x+1)}$$ 5. **Combine the fractions:** $$F(x) = a + \frac{b(x-3) + c(x+1)}{(x+1)(x-3)} = a + \frac{(b+c)x + (c - 3b)}{(x+1)(x-3)}$$ 6. **Final form:** $$F(x) = a + \frac{(b+c)x + (c - 3b)}{x^2 - 2x - 3}$$ This form helps us analyze the function's behavior, asymptotes, and intercepts. **Summary:** The function has vertical asymptotes at $$x = -1$$ and $$x = 3$$, and can be expressed as $$F(x) = a + \frac{(b+c)x + (c - 3b)}{x^2 - 2x - 3}$$ for further analysis.