Let's first rewrite the equation using exponents in terms of 2, 6, and 9:

2^(4x) + 3 6^(2x) - 9 9^(2x) = 0

Now, we know that 6 = 2 * 3 and 9 = 3^2, so we can rewrite the equation further:

2^(4x) + 3 (2 3)^(2x) - 9 (3^2)^(2x) = 0
2^(4x) + 3 2^(2x) 3^(2x) - 9 3^(4x) = 0

Now, let's simplify by using the property that a^(m+n) = a^m * a^n:

2^(4x) + 3 2^(2x) 3^(2x) - 9 (3^2)^(2x) = 0
2^(4x) + 3 2^(2x) 3^(2x) - 9 3^(4x) = 0
2^(4x) + 3 2^(2x) 3^(2x) - 9 3^(4x) = 0
2^(4x) + 3 2^(2x) 3^(2x) - 9 3^(4x) = 0

Now, we can see that we have a common base of 2, so let's rewrite the equation:

2^(4x) + 3 2^(2x) 3^(2x) - 9 3^(4x) = 0
2^(4x) + 3 2^(2x) 3^(2x) - 9 (3^2)^(2x) = 0
2^(4x) + 3 2^(2x) 3^(2x) - 9 * 3^(4x) = 0

This equation can be simplified further using the laws of exponents, but the final solution will depend on what x is equal to.