Let's simplify the given expression step by step:

9x^2 - 7(x + 4)(4 - x) - (1 - 4x)^2 = 15

First, expand the terms inside the parentheses using the distributive property:

9x^2 - 7(4x - x^2 + 16 - 4x) - (1 - 4x)(1 - 4x) = 15

Simplify the terms inside the parentheses further:

9x^2 - 28x + 7(16) - 7(-x^2) - 7(4x) - (1 - 8x + 16x - 16x^2) = 15

Now, continue simplifying the equation:

9x^2 - 28x + 112 + 7x^2 + 28x - (1 - 8x + 16x - 16x^2) = 15

Combine like terms:

9x^2 + 7x^2 + 28x - 28x + 112 - (1 - 8x + 16x - 16x^2) = 15
16x^2 + 112 - (1 - 8x + 16x - 16x^2) = 15

Now, simplify the equation further:

16x^2 + 112 - 1 + 8x - 16x + 16x^2 = 15

Combine like terms:

32x^2 + 8x + 111 = 15

Now, rearrange the equation and simplify it further:

32x^2 + 8x + 111 - 15 = 0
32x^2 + 8x + 96 = 0

Finally, factor the quadratic equation:

8(4x^2 + x + 12) = 0

The solutions for the given equation are not readily apparent. I apologize for the oversight. Let's use formula.
x = (-b ± sqrt(b^2 - 4ac)) / 2a

For the equation 32x^2 + 8x + 96 = 0, a = 32, b = 8, c = 96.

x = (-8 ± sqrt(8^2 - 43296)) / (2*32)
x = (-8 ± sqrt(64 - 1228)) / 64
x = (-8 ± sqrt(-1164)) / 64
x = (-8 ± 34.09i) / 64

Therefore, the solutions for the given equation are complex numbers:

x = (-8 + 34.09i) / 64 or x = (-8 - 34.09i) / 64

These are the simplified solutions for the given expression.