1. **State the problem:** We are given the difference quotient expression for a function $f$: $$f(x + h) - f(x) = -1hx^2 - 5hx + 8h^2x - 1h^2 - 2h^3$$ and we need to find the derivative $f'(x)$. 2. **Recall the definition of the derivative:** $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 3. **Apply the formula:** Divide the given expression by $h$: $$\frac{f(x+h) - f(x)}{h} = \frac{-1hx^2 - 5hx + 8h^2x - 1h^2 - 2h^3}{h}$$ 4. **Simplify by canceling $h$:** $$= \frac{\cancel{h}(-1x^2 - 5x) + h^2(8x) - h^2(1) - 2h^3}{\cancel{h}} = -1x^2 - 5x + 8hx - 1h - 2h^2$$ 5. **Take the limit as $h \to 0$:** Since terms with $h$ and $h^2$ vanish, $$f'(x) = \lim_{h \to 0} \left(-1x^2 - 5x + 8hx - 1h - 2h^2\right) = -1x^2 - 5x$$ 6. **Final answer:** $$\boxed{f'(x) = -x^2 - 5x}$$