1. **State the problem:** Find the difference quotient $$\frac{f(x+h)-f(x)}{h}$$ for the function $$f(x) = x^2 - 3x$$. 2. **Write the formula for the difference quotient:** $$\frac{f(x+h)-f(x)}{h}$$ 3. **Calculate $$f(x+h)$$:** $$f(x+h) = (x+h)^2 - 3(x+h)$$ Expand the square and distribute: $$= x^2 + 2xh + h^2 - 3x - 3h$$ 4. **Calculate $$f(x+h) - f(x)$$:** $$= (x^2 + 2xh + h^2 - 3x - 3h) - (x^2 - 3x)$$ Simplify by canceling terms: $$= \cancel{x^2} + 2xh + h^2 - 3x - 3h - \cancel{x^2} + 3x$$ $$= 2xh + h^2 - 3h$$ 5. **Form the difference quotient:** $$\frac{f(x+h)-f(x)}{h} = \frac{2xh + h^2 - 3h}{h}$$ 6. **Simplify by factoring out $$h$$ in the numerator:** $$= \frac{h(2x + h - 3)}{h}$$ Cancel $$h$$: $$= \cancel{\frac{h}{h}}(2x + h - 3)$$ $$= 2x + h - 3$$ **Final answer:** $$\frac{f(x+h)-f(x)}{h} = 2x + h - 3$$ This expression represents the average rate of change of the function $$f$$ over the interval from $$x$$ to $$x+h$$.