Expanding each side of the equation:

(x+2)^2 = (x+2)(x+2) = x^2 + 4x + 4

(x-4)^2 = (x-4)(x-4) = x^2 - 8x + 16

So the original equation simplifies to:

x^2 + 4x + 4 = x^2 - 8x + 16

Now we can combine like terms and simplify:

4x + 4 = -8x + 16

Adding 8x to both sides:

12x + 4 = 16

Subtracting 4 from both sides:

12x = 12

Dividing by 12 on both sides:

x = 1

Therefore, the solution to the equation is x = 1.