To solve this trigonometric equation, we can use the double angle and difference of angles identities.

First, let's expand the terms using the double angle identity:

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

Now, we can rewrite the equation as:

4(2sin(x)cos(x)) - 3(sin(2x)cos(π/3) - cos(2x)sin(π/3)) = 5

Simplify further:

8sin(x)cos(x) - 3(sin(2x)cos(π/3) - cos(2x)sin(π/3)) = 5

Now, we will use the sum and difference of angles identities to find expressions for sin(2x) and cos(2x):

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

Substitute these expressions into the equation:

8sin(x)cos(x) - 3(2sin(x)cos(x)cos(π/3) - (cos^2(x) - sin^2(x))sin(π/3)) = 5

Now, simplify and solve for x. This will involve using trigonometric identities and manipulating the terms in the equation.