1. **Problem Statement:** Factorize the expression $x^2 y^2 - 4$. 2. **Formula Used:** Recognize this as a difference of squares, which follows the rule: $$a^2 - b^2 = (a - b)(a + b)$$ 3. **Identify Terms:** Here, $a = xy$ and $b = 2$ because: $$x^2 y^2 = (xy)^2 \quad \text{and} \quad 4 = 2^2$$ 4. **Apply the Formula:** Using the difference of squares formula: $$x^2 y^2 - 4 = (xy - 2)(xy + 2)$$ 5. **Explanation:** This factorization works because the product of $(xy - 2)$ and $(xy + 2)$ expands back to $x^2 y^2 - 4$ by the distributive property. 6. **Check Other Options:** - Option B: $(x - y)(x + y)$ is the difference of squares for $x^2 - y^2$, which is not the original expression. - Option C: $(xy - 4)(xy + 4)$ would expand to $x^2 y^2 - 16$, which is incorrect. - Option D: $(x - 2y)(x + 2y)$ expands to $x^2 - 4y^2$, which is different. **Final Answer:** Option A: $(xy - 2)(xy + 2)$