The problem is to simplify the expression $(a+2b)^2(a-2b)^2$.

1. Problem Description

The problem is to simplify the expression $(a+2b)^2(a-2b)^2$.

2. Solution Steps

First, we can rewrite the expression as $[(a+2b)(a-2b)]^2$.

Then, we can use the difference of squares formula:

$(x+y)(x-y) = x^2 - y^2$

In our case, $x = a$ and $y = 2b$.

So, $(a+2b)(a-2b) = a^2 - (2b)^2 = a^2 - 4b^2$.

Now, we substitute this back into the expression:

$[(a+2b)(a-2b)]^2 = (a^2 - 4b^2)^2$.

We then expand the square:

$(a^2 - 4b^2)^2 = (a^2)^2 - 2(a^2)(4b^2) + (4b^2)^2$

$= a^4 - 8a^2b^2 + 16b^4$.

3. Final Answer

$a^4 - 8a^2b^2 + 16b^4$