4tan^4 x + tan^2 x - 3 = sec^2 x (4tan^2 x - 3)
The left side factors into:
(tan² x + 1)(4tan² x - 3) = sec² x (4tan² x - 3)
The (4tan² x - 3) factors cancel each other to leave:
tan² x + 1 = sec² x
This is one of the Pythagorean Identities, so this is proved.
I hope that helps!! :)