I tried the 2022 MAT paper and got all the multiple choice questions correct except this one:
Spoilers below if those reading this want to attempt it
I spent a while looking at the given factorisation and trying to work out how it connects to when n^4+4 is prime but in the end I gave up (I only knew it wasn't (a) because of n=1). The solution is to make a substitution n=m/sqrt(2) which makes sense but I'm not sure I would have spotted that in a reasonable time.
If I wasn't given the factorisation and just asked about n^4+4 then I would have attempted to factorise it and I don't think it's too hard to realise that it's (n^2 + 2n + 2)(n^2 - 2n + 2). Then I'm confident that I would have got the answer from there. I should have tried factorising n^4+4 but I was too focussed on trying to use the given result.
So I'm wondering what the point of giving the n^4+1 factorisation is because it doesn't seem necessary and was a big distraction, for me at least? Is this an "obvious" technique that Oxbridge students would be expected to see?