To solve the equation sin(3x) + sin(x) + 2cos(x) = sin(2x) + 2cos^2(x), we need to simplify both sides of the equation.

Starting with the left side:
sin(3x) + sin(x) + 2cos(x)
Using the trigonometric identity sin(3x) = 3sin(x) - 4sin^3(x), this simplifies to:
3sin(x) - 4sin^3(x) + sin(x) + 2cos(x)
Combining like terms:
4sin(x) - 4sin^3(x) + 2cos(x)

Next, simplifying the right side:
sin(2x) + 2cos^2(x)
Using the trigonometric identities sin(2x) = 2sin(x)cos(x) and cos^2(x) = 1 - sin^2(x), this simplifies to:
2sin(x)cos(x) + 2(1 - sin^2(x))
Expanding and combining like terms:
2sin(x)cos(x) + 2 - 2sin^2(x)

Now, we have:
4sin(x) - 4sin^3(x) + 2cos(x) = 2sin(x)cos(x) + 2 - 2sin^2(x)

Since the equation now only contains sin and cos terms, we need to use trigonometric identities to simplify further. By using the identity 2sin(x)cos(x) = sin(2x), the equation becomes:
4sin(x) - 4sin^3(x) + 2cos(x) = sin(2x) + 2 - 2sin^2(x)

At this point, you may want to experiment with various trigonometric identities to simplify the equation further. With some algebraic manipulation and use of additional identities, you can potentially arrive at a solution that verifies the equality of both sides of the equation.