1. The problem is to find a formula for $\alpha^3 + \beta^3$. 2. Recall the sum of cubes formula: for any two numbers $a$ and $b$, $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ 3. Applying this to $\alpha$ and $\beta$, we get: $$\alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2)$$ 4. This formula is useful because it expresses the sum of cubes as a product of a binomial and a trinomial. 5. To use it, you need to know or calculate $\alpha + \beta$ and $\alpha\beta$ if you want to simplify further. 6. For example, if you know $\alpha + \beta = s$ and $\alpha\beta = p$, then: $$\alpha^2 - \alpha\beta + \beta^2 = (\alpha + \beta)^2 - 3\alpha\beta = s^2 - 3p$$ 7. So the sum of cubes can also be written as: $$\alpha^3 + \beta^3 = s(s^2 - 3p)$$ This completes the explanation and formula for solving $\alpha^3 + \beta^3$.