The problem asks us to factor the expression $x^4 + x^2y^2 + y^4$.

1. Problem Description

The problem asks us to factor the expression $x^4 + x^2y^2 + y^4$.

2. Solution Steps

We can rewrite the given expression by adding and subtracting $x^2y^2$:

$x^4 + x^2y^2 + y^4 = x^4 + 2x^2y^2 + y^4 - x^2y^2$

Now, we can factor the first three terms as a perfect square:

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

So the expression becomes:

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

This is a difference of squares, which can be factored as:

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

In our case, $a = (x^2 + y^2)$ and $b = xy$. Therefore,

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

Rearranging the terms within the parentheses, we get:

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

3. Final Answer

The factored form of the expression $x^4 + x^2y^2 + y^4$ is $(x^2 - xy + y^2)(x^2 + xy + y^2)$.