factor $$ {\displaylines x^4+2x^2y^2+y^4\\ } $$

The given expression is:

\[
x^4 + 2x^2y^2 + y^4
\]

This expression can be recognized as a perfect square trinomial. The general form of a perfect square trinomial is:

\[
a^2 + 2ab + b^2 = (a + b)^2
\]

In the given expression:

- $ a^2 = x^4 $, so $ a = x^2 $
- $ b^2 = y^4 $, so $ b = y^2 $
- $ 2ab = 2x^2y^2 $, which matches the middle term.

Therefore, the given expression can be factored as:

\[
(x^2 + y^2)^2
\]

So, the factorization of $ x^4 + 2x^2y^2 + y^4 $ is:

\[
(x^2 + y^2)^2
\]

Would you like more details or have any questions?

Here are five related questions:
1. How would you factor the expression $ x^4 + y^4 $?
2. Can you factor $ x^4 - 2x^2y^2 + y^4 $ using a similar method?
3. What is the expanded form of $ (x^2 - y^2)^2 $?
4. How would you factor the expression $ x^4 - y^4 $?
5. Can the expression $ x^4 + 4x^2y^2 + 4y^4 $ be factored as a square?

**Tip:** Recognizing patterns like perfect square trinomials can simplify the process of factoring complex expressions.

Learn how to factor x^4 + 2x^2y^2 + y^4 using the perfect square trinomial method with a step-by-step solution. This problem involves recognizing patterns in algebraic expressions and applying the formula (a + b)^2 = a^2 + 2ab + b^2. Suitable for high school students studying algebra.

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