To solve for x in the equation 2sin²x - 5sinxcosx = 3, we can rewrite the equation in terms of sin2x:

2sin²x - 5sinxcosx = 3
2sin²x - 5(1/2)sin2x = 3
2sin²x - 5sin2x/2 = 3
2sin²x - 5sin2x/2 = 3
4sin²x - 5sin2x = 6
4sin²x - 5(2sinxcosx) = 6
4sin²x - 10sinxcosx = 6
4(1 - cos²x) - 10sinxcosx = 6
4 - 4cos²x - 10sinxcosx = 6
-4cos²x - 10sinxcosx = 2
2cos²x + 5sinxcosx = -1

Now we have a new equation in terms of sinx and cosx. We can recognize the Pythagorean identity sin²x + cos²x = 1 and substitute for sin²x:

2(1 - cos²x) + 5sinxcosx = -1
2 - 2cos²x + 5sinxcosx = -1
2 - 2cos²x + 5sinxcosx = -1

Let cosx = u,
2 - 2u² + 5sin(u) = -1
2 - 2u² + 5sqrt(1 - u²) = -1
2 - 2u² + 5sqrt(1 - u²) = -1

Solving this equation may be difficult since it does not have a simple solution in terms of elementary functions. A numerical method like Newton's method or a graphing calculator may be needed to approximate the value of u and find possible values of x.