To solve the equation sin(2x) + 4sin^2(x) = 2cos^2(x), we can use the double angle identity for sin(2x) and the Pythagorean identity sin^2(x) + cos^2(x) = 1.

sin(2x) = 2sin(x)cos(x)

Now substitute these identities into the equation:

2sin(x)cos(x) + 4sin^2(x) = 2(1 - sin^2(x))

2sin(x)cos(x) + 4sin^2(x) = 2 - 2sin^2(x)

Rearrange the terms:

6sin^2(x) + 2sin(x)cos(x) - 2 = 0

Now we have a quadratic equation in terms of sin(x). To solve for sin(x), we can use the quadratic formula:

sin(x) = [-2cos(x) ± √(4cos^2(x) + 48)] / 12

Now, we need to substitute back sin(x) into the equation to find the possible values of x.