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. **Recall the formula:** The difference quotient represents the average rate of change of the function over the interval from $a$ to $a+h$. It is given by $$\frac{f(a+h) - f(a)}{h}$$. 3. **Calculate $f(a+h)$:** Substitute $x = a+h$ into $f(x)$: $$f(a+h) = 2(a+h)^2 - 5(a+h) + 1$$ Expand the square: $$= 2(a^2 + 2ah + h^2) - 5a - 5h + 1$$ Distribute: $$= 2a^2 + 4ah + 2h^2 - 5a - 5h + 1$$ 4. **Calculate $f(a)$:** Substitute $x = a$: $$f(a) = 2a^2 - 5a + 1$$ 5. **Form the numerator $f(a+h) - f(a)$:** $$= (2a^2 + 4ah + 2h^2 - 5a - 5h + 1) - (2a^2 - 5a + 1)$$ Simplify by subtracting: $$= 2a^2 + 4ah + 2h^2 - 5a - 5h + 1 - 2a^2 + 5a - 1$$ Combine like terms: $$= 4ah + 2h^2 - 5h$$ 6. **Divide by $h$:** $$\frac{4ah + 2h^2 - 5h}{h}$$ Factor $h$ from numerator: $$= \frac{h(4a + 2h - 5)}{h}$$ Cancel $h$ (since $h \neq 0$): $$= 4a + 2h - 5$$ 7. **Final answer:** $$\boxed{4a + 2h - 5}$$ This expression gives the average rate of change of $f(x)$ from $x=a$ to $x=a+h$.