1. **State the problem:** We need to find all x-intercepts of the function $$f(x) = -2x^4 + 2x$$. The x-intercepts occur where $$f(x) = 0$$. 2. **Set the function equal to zero:** $$-2x^4 + 2x = 0$$ 3. **Factor the equation:** First, factor out the common factor $$2x$$: $$2x(-x^3 + 1) = 0$$ 4. **Apply the zero product property:** Set each factor equal to zero: $$2x = 0$$ or $$-x^3 + 1 = 0$$ 5. **Solve each equation:** - For $$2x = 0$$: $$\cancel{2}x = 0 \Rightarrow x = 0$$ - For $$-x^3 + 1 = 0$$: $$-x^3 = -1$$ $$x^3 = 1$$ $$x = \sqrt[3]{1} = 1$$ 6. **List all x-intercepts:** The x-intercepts are $$x = 0$$ and $$x = 1$$. **Final answer:** $$\boxed{0, 1}$$