Factoring the Expression (x^2 + y^2  5)^2  4x^2y^2  16xy  16
This expression appears complex, but we can simplify it through careful factoring. Here's a stepbystep breakdown:
1. Recognizing Patterns
 Difference of Squares: Notice that the first term is a perfect square: (x^2 + y^2  5)^2.
 Perfect Square Trinomial: The remaining terms (4x^2y^2  16xy  16) can be rearranged and factored into a perfect square trinomial.
2. Applying the Patterns
 Difference of Squares: (a^2  b^2) = (a + b)(a  b)
 Perfect Square Trinomial: (a^2 + 2ab + b^2) = (a + b)^2
Let's apply these:

Rewrite the expression: (x^2 + y^2  5)^2  (2xy + 4)^2

Factor using the difference of squares: [(x^2 + y^2  5) + (2xy + 4)][(x^2 + y^2  5)  (2xy + 4)]

Simplify: (x^2 + y^2 + 2xy  1)(x^2 + y^2  2xy  9)

Factor the remaining terms: [(x + y)^2  1][(x  y)^2  9]

Apply difference of squares again: [(x + y + 1)(x + y  1)][(x  y + 3)(x  y  3)]
Final Result
The factored form of the expression (x^2 + y^2  5)^2  4x^2y^2  16xy  16 is:
(x + y + 1)(x + y  1)(x  y + 3)(x  y  3)
Key Takeaways
 Recognize patterns: Identifying patterns like the difference of squares and perfect square trinomials is crucial for simplifying expressions.
 Break it down: Factor the expression step by step, using the appropriate pattern at each stage.
 Simplify and refine: Once factored, check for further simplification opportunities.
This exercise demonstrates how factoring can transform complex expressions into more manageable and understandable forms.