To solve the quadratic equation 2lg x^2 - lg^2(-x) - 4 = 0, we need to simplify the equation.

First, let's simplify the equation:

2lg x^2 - lg^2(-x) - 4 = 0
2lg(x^2) - lg^2(-x) - 4 = 0
lg(x^4) - lg^2(-x) - 4 = 0
lg(x^4) + lg(x^2) - 4 = 0

Now, we can combine the logarithms using the properties of logarithms:

lg(x^4 * x^2) - 4 = 0
lg(x^6) - 4 = 0

Next, we can rewrite the equation in exponential form:

x^6 = 10^4
x^6 = 10000

Take the 6th root of both sides to solve for x:

x = ±(10000)^(1/6)
x = ±(10)^(4/6)
x = ±10^(2/3)
x = ±10^(0.6667)
x ≈ ±4.64

Therefore, the solutions to the equation are x ≈ 4.64 and x ≈ -4.64.