Certainly! Let's break down the given expression [tex]\( x^4 + 2x^2y^2 + y^4 \)[/tex] in a detailed step-by-step manner.
1. Identify the terms in the expression:
The expression is:
[tex]\[
x^4 + 2x^2y^2 + y^4
\][/tex]
2. Recognize the structure:
Notice that this expression can be seen as a polynomial in terms of powers of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Specifically, observe that:
[tex]\[
x^4, \quad 2x^2y^2, \quad y^4
\][/tex]
3. Factorization through pattern recognition:
We see that the structure of the polynomial resembles a common algebraic identity. The general identity for the square of a binomial sum is given by:
[tex]\[
(a^2 + b^2)^2 = a^4 + 2a^2b^2 + b^4
\][/tex]
In our case, identify:
[tex]\[
a = x^2 \quad \text{and} \quad b = y^2
\][/tex]
4. Rewrite using the binomial identity:
So, if we set [tex]\( a = x^2 \)[/tex] and [tex]\( b = y^2 \)[/tex] in the identity, our expression becomes:
[tex]\[
(x^2)^2 + 2(x^2)(y^2) + (y^2)^2
\][/tex]
5. Simplify the binomial expression:
Thus, we can write:
[tex]\[
x^4 + 2x^2y^2 + y^4 = (x^2 + y^2)^2
\][/tex]
Hence, our given expression:
[tex]\[
x^4 + 2x^2y^2 + y^4
\][/tex]
simplifies to:
[tex]\[
(x^2 + y^2)^2
\][/tex]
### Final Answer:
The given expression [tex]\( x^4 + 2x^2y^2 + y^4 \)[/tex] simplifies to:
[tex]\[
(x^2 + y^2)^2
\][/tex]
