1. **State the problem:** We are given the function $f(x) = 2x^2 - 5x + 1$ and need to evaluate the difference quotient $$\frac{f(a+h) - f(a)}{h}$$ where $h \neq 0$. 2. **Find $f(a+h)$:** Substitute $x = a+h$ into $f(x)$: $$f(a+h) = 2(a+h)^2 - 5(a+h) + 1$$ Expand the square: $$(a+h)^2 = a^2 + 2ah + h^2$$ So, $$f(a+h) = 2(a^2 + 2ah + h^2) - 5a - 5h + 1 = 2a^2 + 4ah + 2h^2 - 5a - 5h + 1$$ 3. **Find $f(a)$:** Substitute $x = a$: $$f(a) = 2a^2 - 5a + 1$$ 4. **Compute the numerator $f(a+h) - f(a)$:** $$f(a+h) - f(a) = (2a^2 + 4ah + 2h^2 - 5a - 5h + 1) - (2a^2 - 5a + 1)$$ Simplify by canceling terms: $$= 2a^2 + 4ah + 2h^2 - 5a - 5h + 1 - 2a^2 + 5a - 1 = 4ah + 2h^2 - 5h$$ 5. **Divide by $h$ (with $h \neq 0$):** $$\frac{f(a+h) - f(a)}{h} = \frac{4ah + 2h^2 - 5h}{h} = 4a + 2h - 5$$ **Final answer:** $$\boxed{4a + 2h - 5}$$