Let's simplify the given expression:

(3x^2 - 4)^2 - 4(3x^2 - 4) - 5 = 0

Expanding the terms inside the parentheses:

= (9x^4 - 24x^2 + 16) - (12x^2 - 16) - 5

Distribute the negative sign inside the parentheses:

= 9x^4 - 24x^2 + 16 - 12x^2 + 16 - 5

Combine like terms:

= 9x^4 - 36x^2 + 27

Now, the equation becomes:

9x^4 - 36x^2 + 27 = 0

This is a quadratic equation in terms of x^2. To solve for x^2, we can set it equal to zero and factor:

9x^4 - 36x^2 + 27 = 0

Divide the equation by 9 to simplify:

x^4 - 4x^2 + 3 = 0

Now, we can factor this as a quadratic equation:

(x^2 - 3)(x^2 - 1) = 0

Setting each factor to zero gives us the solutions:

x^2 - 3 = 0 or x^2 - 1 = 0

x^2 = 3 or x^2 = 1

Taking the square root of both sides gives us:

x = ±√3 or x = ±1

Therefore, the solutions for the given equation are x = ±√3 and x = ±1.