To solve the equation 2x + √(4|x| - 1) = 0, we need to isolate x.

First, let's isolate the square root term by subtracting 2x from both sides:

√(4|x| - 1) = -2x

Next, square both sides of the equation to eliminate the square root:

(√(4|x| - 1))^2 = (-2x)^2

4|x| - 1 = 4x^2

Now, we can split the equation into two cases based on the absolute value:

Case 1: |x| = x

In this case, we can rewrite the equation as:

4x - 1 = 4x^2

Rearranging the terms:

4x^2 - 4x + 1 = 0

This is a quadratic equation. We can solve it by factoring or using the quadratic formula. However, upon inspection, we can see that this quadratic equation does not have real solutions. Therefore, there are no solutions in this case.

Case 2: |x| = -x

In this case, we can rewrite the equation as:

4(-x) - 1 = 4x^2

Simplifying:

-4x - 1 = 4x^2

Rearranging the terms:

4x^2 + 4x + 1 = 0

This is a quadratic equation. We can solve it by factoring or using the quadratic formula. However, upon inspection, we can see that this quadratic equation does not have real solutions. Therefore, there are no solutions in this case.

In conclusion, the equation 2x + √(4|x| - 1) = 0 has no real solutions.