1. **State the problem:** Simplify the expression $\alpha\beta^2 + \alpha^2\beta$. 2. **Identify common factors:** Both terms contain $\alpha$ and $\beta$. 3. **Factor out the common terms:** $$\alpha\beta^2 + \alpha^2\beta = \alpha\beta(\beta + \alpha)$$ 4. **Explanation:** We took out the smallest powers of $\alpha$ and $\beta$ common to both terms, which are $\alpha^1$ and $\beta^1$, leaving $\beta$ in the first term and $\alpha$ in the second inside the parentheses. 5. **Final answer:** $$\alpha\beta(\alpha + \beta)$$