We can simplify this expression using the identity:

sin^2x + cos^2x = 1

Let's start by rearranging the terms in the given equation:

sin^2x - 4sinxcosx + 3cos^2x = 0

Now, we can rewrite the middle term as:

-4sinxcosx = -2(2sinxcosx) = -2sin2x

Using this substitution, the equation becomes:

sin^2x - 2sin2x + 3cos^2x = 0

Now, we can use the identity sin^2x + cos^2x = 1 to rewrite the cosine term:

sin^2x - 2sin2x + 3(1 - sin^2x) = 0

Expanding the brackets and simplifying, we get:

4sin^2x - 2sin2x - 3 = 0

We can factor this expression as:

(2sinx - 3)(2sinx + 1) = 0

Therefore, the solutions are:

2sinx - 3 = 0 or 2sinx + 1 = 0

Solving for sinx in each case, we get:

sinx = 3/2 (not possible, as sinx can only be between -1 and 1)

or

sinx = -1/2

Therefore, the solutions to the original equation are:

x = (7?/6) + 2?n or x = (11?/6) + 2?n, where n is an integer.