The problem is to prove that the quintic $$x^5+10x^4+15x^3+15x^2-10x+1$$ is irreducible in the rationals.
I don't have much knowledge in group theory, and certainly not in Galois theory, and I'm pretty sure this problem can be solved without those tools.
I know about Eisenstein's criterion, but it cannot be applied to this particular quintic because $5$ does not divide the constant term. If we somehow manipulate the polynomial so that $5$ divides the constant term, we still have to make sure that $25$ doesn't.
So is there any other easy ("elementary") way to solve this?