x4 - 5x2y2 + 4y4

(x2)2 - 5(x2)(y2) + 4(y2)2

Let x2 = u and y2 = v

Then the problem becomes

u2 - 5uv + 4v2

We think of two integers whose product is +4 and whose sum is -5,
They are -4 and -1, so that factors as

(u - 4v)(u - v)

(x2 - 4y2)(x2 - y2)

Then each of those factors as the difference of squares:

(x - 2y)(x + 2y)(x - y)(x + y)

Edwin