that's what I did originally, but I ended up with 5 terms, and trying to simplify them all in terms of α + β and αβ became too much of a headache I'm pretty sure there's a way to do it using α^2 + β^2 but I've got no idea what it is
that's what I did originally, but I ended up with 5 terms, and trying to simplify them all in terms of α + β and αβ became too much of a headache I'm pretty sure there's a way to do it using α^2 + β^2 but I've got no idea what it is
You also have to use (a+b)^2, but not yet.
When you expans (a+b)^4 you should be able to write it in terms of a^4+b^4, ab and another term which you need another trick for.
You made a typo in the first line, otherwise fine. You have now reduced the problem to finding a^2+b^2, which you can do using the same method (consider (a+b)^2).