To solve the equation 4sinXcosX + 3cos^2 X = 0, we can use trigonometric identities to simplify the expression.

First, we can rewrite sinXcosX as (1/2)sin2X using the double angle identity. Therefore, the equation becomes:

2sin2X + 3cos^2 X = 0

Next, we can use the Pythagorean identity sin^2 X + cos^2 X = 1 to write sin2X in terms of cosX:

2(1 - cos^2 X) + 3cos^2 X = 0
2 - 2cos^2 X + 3cos^2 X = 0
2 + cos^2 X = 0

Now we have a simple equation to solve for cosX:

cos^2 X = -2

Since square of a real number cannot be negative, there are no real solutions for this equation. The equation 4sinXcosX + 3cos^2 X = 0 has no real solutions.