To solve this trigonometric equation, we can apply various trigonometric identities to simplify and manipulate the expression.

Given equation: 3cos^2(x) = 4sin(x)cos(x) - sin^2(x)

We know that sin^2(x) + cos^2(x) = 1 (Pythagorean identity)

Rewrite the given equation using sin^2(x) + cos^2(x) = 1:
3(1 - sin^2(x)) = 4sin(x)cos(x) - sin^2(x)

Distribute the 3 on left side:
3 - 3sin^2(x) = 4sin(x)cos(x) - sin^2(x)

Rearrange the terms:
3 - 3sin^2(x) + sin^2(x) = 4sin(x)cos(x)
3 - 2sin^2(x) = 4sin(x)cos(x)

Replace 2sin^2(x) with 1 - cos^2(x) (from Pythagorean identity):
3 - 1 + cos^2(x) = 4sin(x)cos(x)
2 + cos^2(x) = 4sin(x)cos(x)

Replace cos^2(x) with 1 - sin^2(x) (from Pythagorean identity):
2 + 1 - sin^2(x) = 4sin(x)cos(x)
3 - sin^2(x) = 4sin(x)cos(x)

Rearranging the terms:
3 = 4sin(x)cos(x) + sin^2(x)

Therefore, the solution to the original equation 3cos^2(x) = 4sin(x)cos(x) - sin^2(x) is 3 = 4sin(x)cos(x) + sin^2(x).