1. **State the problem:** Factor the expression $$16x^2y^8 - 4$$. 2. **Identify common factors:** Both terms have a common factor of 4. 3. **Factor out the greatest common factor (GCF):** $$16x^2y^8 - 4 = \cancel{4} \times 4x^2y^8 - \cancel{4} \times 1 = 4(4x^2y^8 - 1)$$ 4. **Recognize difference of squares:** The expression inside the parentheses is a difference of squares since $$4x^2y^8 = (2xy^4)^2$$ and $$1 = 1^2$$. 5. **Apply difference of squares formula:** $$a^2 - b^2 = (a - b)(a + b)$$ 6. **Factor the expression:** $$4(4x^2y^8 - 1) = 4\big((2xy^4)^2 - 1^2\big) = 4(2xy^4 - 1)(2xy^4 + 1)$$ **Final answer:** $$4(2xy^4 - 1)(2xy^4 + 1)$$