1. **State the problem:** We are given the difference quotient expression for a function $f$: $$f(x + h) - f(x) = -2hx^2 + 5hx + 7h^2x + 5h^2 + 3h^3$$ and we need to find the derivative $f'(x)$. 2. **Recall the definition of the derivative:** $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ This means we divide the given expression by $h$ and then take the limit as $h$ approaches 0. 3. **Divide the expression by $h$:** $$\frac{f(x+h) - f(x)}{h} = \frac{-2hx^2 + 5hx + 7h^2x + 5h^2 + 3h^3}{h}$$ Cancel $h$ in numerator and denominator: $$= \frac{\cancel{h}(-2x^2 + 5x) + h^2(7x + 5) + 3h^3}{\cancel{h}}$$ which simplifies to: $$= -2x^2 + 5x + 7hx + 5h + 3h^2$$ 4. **Take the limit as $h \to 0$:** $$f'(x) = \lim_{h \to 0} \left(-2x^2 + 5x + 7hx + 5h + 3h^2\right) = -2x^2 + 5x$$ because all terms with $h$ vanish. 5. **Final answer:** $$f'(x) = -2x^2 + 5x$$