What is the correct identity for the sum of cubes and how can it be used to solve alpha3 + beta3?

The correct identity for the sum of cubes is:

$$\alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2)$$

We also know that $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$.
Substituting this into the sum of cubes identity gives:

$$\alpha^3 + \beta^3 = (\alpha + \beta)((\alpha + \beta)^2 - 3\alpha\beta)$$