To tackle the expression [tex]\(\left((x + y)^2 2 x y\right)^2\)[/tex] comprehensively, we will go through the steps to simplify it step by step.
1. Expression Analysis and Interpretation:
Initially, the given expression might seem ambiguous, so clarifying it is crucial. Let's consider the expression as: [tex]\(\left((x + y)^2 \cdot 2 \cdot x \cdot y\right)^2\)[/tex].
2. Evaluate the Inner Part:
We first simplify the inner part before squaring it.
- The inner part is [tex]\((x + y)^2 \cdot 2 \cdot x \cdot y\)[/tex].
3. Simplify the Inner Part [tex]\((x + y)^2 \cdot 2 \cdot x \cdot y\)[/tex]:
- Start by expanding [tex]\((x + y)^2\)[/tex]:
[tex]\[
(x + y)^2 = x^2 + 2xy + y^2
\][/tex]
- Now multiply this expanded polynomial by [tex]\(2xy\)[/tex]:
[tex]\[
(x^2 + 2xy + y^2) \cdot 2xy
\][/tex]
- Distribute [tex]\(2xy\)[/tex] through each term in the polynomial:
[tex]\[
2xy \cdot x^2 + 2xy \cdot 2xy + 2xy \cdot y^2
\][/tex]
- Calculating each term:
[tex]\[
2x^3y + 4x^2y^2 + 2xy^3
\][/tex]
- Therefore, the inner expression simplifies to:
[tex]\[
2xy(x^2 + 2xy + y^2)
\][/tex]
4. Squaring the Simplified Inner Part:
- Now we take this entire expression and square it:
[tex]\[
\left(2xy(x^2 + 2xy + y^2)\right)^2
\][/tex]
- Recognizing a simpler way to maintain clarity, recall earlier theoretical conclusions that account for how certain transformations simplify the outer squaring operation.
5. Simplifying Squared Expression:
- When squared, each component and operation will be squared including the distributive polynomial:
[tex]\[
\left(2xy(x^2 + 2xy + y^2) \right)^2
\][/tex]
- Using properties of exponents and distributive rules, we obtain:
[tex]\[
(2^2)(x^2)^2(y^2)^2(x^2 + 2xy + y^2)^2
\][/tex]
- This evaluates to:
[tex]\[
4x^2y^2(x + y)^4
\][/tex]
Putting it all together, the simplified form of the original expression [tex]\(\left((x + y)^2 2 x y\right)^2\)[/tex] is:
[tex]\[
4x^2 y^2 (x + y)^4
\][/tex]
Therefore, our final answer is:
[tex]\[
\boxed{4 x^2 y^2 (x + y)^4}
\][/tex]
