To solve the equation 4sin²x - sinx - 2cos²x = 0, we can use the trigonometric identity sin²x + cos²x = 1.

Rearranging the equation, we get:

4sin²x - sinx - 2(1 - sin²x) = 0
4sin²x - sinx - 2 + 2sin²x = 0
6sin²x - sinx - 2 = 0

Now, let's us substitute sinx = t and solve the quadratic equation 6t² - t - 2 = 0 by factoring or using the quadratic formula.

6t² - t - 2 = 0
(3t + 2)(2t - 1) = 0
t = -2/3 or t = 1/2

Now, recall that t = sinx, so:

sinx = -2/3 or sinx = 1/2

Thus, the solutions to the equation are x = arcsin(-2/3) and x = arcsin(1/2).