The problem is to simplify the expression $(x^2 + 2xy + y^2) - 4z^4$.

1. Problem Description

The problem is to simplify the expression $(x^2 + 2xy + y^2) - 4z^4$.

2. Solution Steps

First, we recognize the expression inside the parentheses as a perfect square trinomial.

$(a+b)^2 = a^2 + 2ab + b^2$

Using this formula, we can simplify the expression inside the parentheses:

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

So the expression becomes:

$(x+y)^2 - 4z^4$

Next, we recognize that $4z^4$ can be written as $(2z^2)^2$.

So the expression becomes:

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

Now, we have a difference of squares. We can use the difference of squares formula:

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

In our case, $a = (x+y)$ and $b = 2z^2$. Applying the difference of squares formula gives:

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

3. Final Answer

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