Let's simplify the equation step by step.

First, let's express all the terms in powers of 4:

12 4^(x^2) - 2 4^(x^2 + 2) + 16 4^(x^2 - 2) = -19 4^(6x + 2)

Rewrite the equation:

12 4^(x^2) - 2 4^2 4^(x^2) + 16 4^(-2) 4^(x^2) = -19 4^(6x) * 4^2

Now simplify the exponents:

12 4^(x^2) - 2 16 4^(x^2) + 16 (1/16) 4^(x^2) = -19 16 * 4^(6x)

Simplify further:

12 4^(x^2) - 32 4^(x^2) + 4^(x^2) = -304 * 4^(6x)

Combine the terms:

-20 4^(x^2) = -304 4^(6x)

Divide both sides by -20:

4^(x^2) = 15.2 * 4^(6x)

Now both sides are in terms of powers of 4, but solving the equation depends on what you are looking for (an exact solution, or if approximate values are acceptable).