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$sin(2x)cos(\pi /3)-cos(2x)sin(\pi /3)$ = 5

Simplify further:

8sin$x$cos$x$ - 3$sin(2x)cos(\pi /3)-cos(2x)sin(\pi /3)$ = 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$2sin(x)cos(x)cos(\pi /3)-(co{s}^2(x)-si{n}^2(x))sin(\pi /3)$ = 5

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