Expanding the Expression (x²  2xy + y²)(x  2y)
This article explores the process of expanding the given expression: (x²  2xy + y²)(x  2y).
Recognizing Patterns
Before diving into the expansion, let's observe the given expression. The first factor (x²  2xy + y²) is a perfect square trinomial, specifically (x  y)². The second factor (x  2y) is a simple binomial.
Expanding using the Distributive Property
We can expand the expression using the distributive property:

Multiply the first term of the first factor (x²) by each term in the second factor:
 x² * x = x³
 x² * (2y) = 2x²y

Multiply the second term of the first factor (2xy) by each term in the second factor:
 2xy * x = 2x²y
 2xy * (2y) = 4xy²

Multiply the third term of the first factor (y²) by each term in the second factor:
 y² * x = xy²
 y² * (2y) = 2y³

Combine all the terms: x³  2x²y  2x²y + 4xy² + xy²  2y³

Simplify by combining like terms: x³  4x²y + 5xy²  2y³
Conclusion
Therefore, the expanded form of (x²  2xy + y²)(x  2y) is x³  4x²y + 5xy²  2y³. This process showcases the application of the distributive property and recognizing patterns to simplify algebraic expressions.