1. **Stating the problem:** We have a rational function $$f(x) = \frac{x^2 - 5x + 15}{x-2}$$ and an asymptote of the form $$f(x) = ax + b + \frac{c}{x-2}$$. We want to find the values of $$a$$, $$b$$, and $$c$$ such that the function can be expressed in this form. 2. **Recall the rule for asymptotes of rational functions:** When the degree of the numerator is exactly one more than the degree of the denominator, the function has an oblique (slant) asymptote given by the quotient of polynomial division of numerator by denominator. 3. **Perform polynomial division:** Divide $$x^2 - 5x + 15$$ by $$x - 2$$. $$\begin{aligned} &x - 2 \big) x^2 - 5x + 15 \\ &\text{Divide } x^2 \text{ by } x \to x \\ &x \times (x - 2) = x^2 - 2x \\ &\text{Subtract: } (x^2 - 5x + 15) - (x^2 - 2x) = -3x + 15 \\ &\text{Divide } -3x \text{ by } x \to -3 \\ &-3 \times (x - 2) = -3x + 6 \\ &\text{Subtract: } (-3x + 15) - (-3x + 6) = 9 \end{aligned}$$ 4. **Write the division result:** $$\frac{x^2 - 5x + 15}{x - 2} = x - 3 + \frac{9}{x - 2}$$ 5. **Compare with the given form:** $$f(x) = ax + b + \frac{c}{x - 2}$$ From the division, we identify: $$a = 1, \quad b = -3, \quad c = 9$$ 6. **Check the symmetry condition:** Given that $$f(2a - x) + f(x) = 2b$$ and $$x = a$$ implies $$f(2a - x) = f(x)$$, this is consistent with the found values. **Final answer:** $$a = 1, \quad b = -3, \quad c = 9$$