Factoring the expression a^4 + 2a^2b^2 + b^4 requires us to look for common factors among the terms. In this case, we can notice that all the terms are perfect squares: a^4 is the square of a^2, 2a^2b^2 is the square of ab, and b^4 is the square of b^2.

We can rewrite the expression as follows:

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

Now let's consider this expression as a quadratic trinomial with respect to a^2. We can see that the terms follow the pattern of a perfect square trinomial:

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

So, we have factored the expression as (a^2 + b^2)^2.

To summarize:

Factoring a^4 + 2a^2b^2 + b^4:
1. Rewrite the expression as a^4 + 2a^2b^2 + b^4.
2. Notice that all the terms are perfect squares: a^4, 2a^2b^2, and b^4.
3. Write the expression using the pattern of a perfect square trinomial: (a^2)^2 + 2(a^2)(b^2) + (b^2)^2.
4. Recognize that this is a perfect square trinomial and factor it as (a^2 + b^2)^2.

Therefore, the factored form of the expression a^4 + 2a^2b^2 + b^4 is (a^2 + b^2)^2.