1. The problem is to simplify the expression $\alpha^2 - \beta^2$. 2. We use the difference of squares formula: $$a^2 - b^2 = (a - b)(a + b)$$ 3. Applying this formula, we get: $$\alpha^2 - \beta^2 = (\alpha - \beta)(\alpha + \beta)$$ 4. This factorization shows the expression as a product of two binomials. 5. This is the simplest form unless values for $\alpha$ and $\beta$ are given. Final answer: $$(\alpha - \beta)(\alpha + \beta)$$