To solve for x in the equation √3 sin 4x + cos4x = 2, we can use the double angle identity for sine and cosine functions.

sin 2x = 2sin x cos x
cos 2x = cos^2 x - sin^2 x

Now let's rewrite the equation using the double angle identities:

√3(2sin 2x cos 2x) + (cos^2 2x - sin^2 2x) = 2

Expanding, we get:

2√3 sin 2x cos 2x + cos^2 2x - sin^2 2x = 2

Now, let's substitute the double angle identities for sin 2x and cos 2x:

2√3 (2sin x cos x)(cos^2 x - sin^2 x) + (cos^2 x - sin^2 x)^2 = 2

Expanding further:

8√3 sin x cos x cos^2 x - 8√3 sin x cos x sin^2 x + cos^4 x - 2cos^2 x sin^2 x + sin^4 x = 2

At this point, we have a complicated expression that may not easily simplify to solve for x. However, you can try to simplify it further by applying additional trigonometric identities, factoring, or other methods.