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The given expression is a rational algebraic fraction:

\[
\frac{2x^3 + 6x^2 + 6xy^2 + 2y^3}{x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4}
\]

This resembles the expanded form of a binomial raised to a power, specifically the expansion of \((x + y)^4\) in the denominator and a simplified form in the numerator. Let's analyze it step by step:

### Denominator:
The denominator matches the expansion of \((x + y)^4\):
\[
(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
\]

### Numerator:
The numerator can be factored as:
\[
2(x^3 + 3x^2 + 3xy^2 + y^3) = 2(x + y)^3
\]

### Simplified Expression:
Now, we can rewrite the expression as:
\[
\frac{2(x + y)^3}{(x + y)^4}
\]

By simplifying the common factors, we get:
\[
\frac{2}{x + y}
\]

### Final Answer:
The simplified form of the given expression is:
\[
\frac{2}{x + y}
\]

Would you like further explanation on this, or do you have additional questions? Here are a few questions to expand your understanding:

1. How can you derive the binomial expansion formula for \((x + y)^n\)?
2. What happens if \(x + y = 0\) in this context?
3. Can this simplification apply to similar binomial expansions for other powers?
4. How does factoring help simplify complex rational expressions like this?
5. What is the significance of recognizing patterns in polynomial expansions?

**Tip:** Always check if a complex expression can be rewritten in terms of familiar patterns like binomial expansions or common factorizations to simplify it quickly.

Simplify the rational expression: (2x^3 + 6x^2 + 6xy^2 + 2y^3) / (x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4).

Learn how to simplify a rational algebraic expression by identifying binomial expansions and canceling common factors. This step-by-step solution demonstrates the use of the binomial theorem to simplify the fraction (2x^3 + 6x^2 + 6xy^2 + 2y^3) / (x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4), making it ideal for high school students studying algebra.

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