Simplifying the Expression (x+2y)(x2y)(x^2+4y^2)
This expression involves a combination of multiplication and factoring, and it can be simplified using the difference of squares pattern. Here's how:
Understanding the Difference of Squares Pattern
The difference of squares pattern states: (a + b)(a  b) = a²  b²
Applying the Pattern

Focus on the first two factors: (x + 2y)(x  2y)
 This perfectly matches the difference of squares pattern, where:
 a = x
 b = 2y
 This perfectly matches the difference of squares pattern, where:

Apply the pattern:
 (x + 2y)(x  2y) = x²  (2y)² = x²  4y²

Multiply the result by the third factor: (x²  4y²)(x² + 4y²)
 Now we have another difference of squares pattern:
 a = x²
 b = 4y²
 Now we have another difference of squares pattern:

Apply the pattern again:
 (x²  4y²)(x² + 4y²) = (x²)²  (4y²)² = x⁴  16y⁴
Final Result
Therefore, the simplified form of the expression (x + 2y)(x  2y)(x² + 4y²) is x⁴  16y⁴.