To solve this equation, we can first rewrite it in terms of sin(2x) using the identity sin(2x) = 2sin(x)cos(x).

So, 2sin(x)cos(x) + 5cos^2(x) = 4 can be rewritten as 2sin(x)cos(x) + 5cos^2(x) = 4.

Now, we substitute sin(2x) = 2sin(x)cos(x) into the equation, we get:

sin(2x) + 5cos^2(x) = 4.

Next, we can use the Pythagorean identity cos^2(x) = 1 - sin^2(x) to rewrite the equation as:

sin(2x) + 5(1 - sin^2(x)) = 4.

Solving for sin(2x), we get:

sin(2x) = 4 - 5 + 5sin^2(x).
sin(2x) = 5sin^2(x) - 1.

This is a quadratic equation in terms of sin(x), which can be solved by setting it equal to zero and factoring or using the quadratic formula.