To simplify the given expression, we can manipulate the trigonometric identities and equations.

Given: sin^2x - 3cos^2x + 2sinx + cosx = 0

We know that sin^2x + cos^2x = 1

Rewriting the given expression by substituting cos^2x = 1 - sin^2x, we get:

sin^2x - 3(1 - sin^2x) + 2sinx + cosx = 0
sin^2x - 3 + 3sin^2x + 2sinx + cosx = 0
4sin^2x + 2sinx + cosx - 3 = 0

Now, we can substitute sinx = t to simplify the expression:

4t^2 + 2t + √(1 - t^2) - 3 = 0

This is the simplified form of the given trigonometric expression.