x^2 + 1/x^2 - 2

is the same as :

x^4 - 2x^2 + 1

after you multiply every term by x^2.

Then that is just a perfect square so when youfactor it you get

(x^2-1)(x^2-1)

which is two differnece of squares which gives:
(x+1)(x-1)(x+1)(x-1)

Hope that helps.