A Quadratic in Disguise

x^4 - 5x^2 + 4 has degree 4, but look at its powers: x^4 is (x^2)^2. Write u for x^2 and it becomes u^2 - 5u + 4, an ordinary quadratic, which factors as (u - 1)(u - 4).

Put x^2 back in for u: (x^2 - 1)(x^2 - 4). Both brackets are differences of squares, so keep going:

x^4 - 5x^2 + 4 = (x + 1)(x - 1)(x + 2)(x - 2)

The trick works whenever the powers are x^4, x^2 and a number: the top power is double the middle one, so the middle power plays the part of x.