1. **State the problem:** Simplify the expression $2\omega + 2\omega^2$ where $\omega$ is a complex cube root of unity. 2. **Recall the properties of cube roots of unity:** The cube roots of unity satisfy the equation $$\omega^3 = 1$$ and the important identity $$1 + \omega + \omega^2 = 0.$$ From this, we get $$\omega^2 = -1 - \omega.$$ 3. **Substitute $\omega^2$ in the expression:** $$2\omega + 2\omega^2 = 2\omega + 2(-1 - \omega) = 2\omega - 2 - 2\omega.$$ 4. **Simplify the expression:** $$2\omega - 2 - 2\omega = -2.$$ 5. **Final answer:** $$2\omega + 2\omega^2 = -2.$$