1. **Problem statement:** We are asked to perform a Kurvendiskussion (curve discussion) for the function $$f(x) = (x - 1) \cdot (x + 2)^2$$ starting with expanding (Ausklammern) the expression. 2. **Expand the function:** Use the distributive property to expand $$f(x)$$. $$f(x) = (x - 1)(x + 2)^2 = (x - 1)(x^2 + 4x + 4)$$ 3. **Multiply out:** $$f(x) = x(x^2 + 4x + 4) - 1(x^2 + 4x + 4) = x^3 + 4x^2 + 4x - x^2 - 4x - 4$$ 4. **Simplify:** $$f(x) = x^3 + (4x^2 - x^2) + (4x - 4x) - 4 = x^3 + 3x^2 - 4$$ 5. **Final expanded form:** $$\boxed{f(x) = x^3 + 3x^2 - 4}$$ This is the starting point for the full Kurvendiskussion. q_count is 1 because only one function is analyzed.