1. **State the problem:** Simplify the expression $$(x^2 - 4y)^2 - (x^2 + 4y)^2$$. 2. **Recall the formula:** This expression is a difference of squares, which follows the rule $$a^2 - b^2 = (a - b)(a + b)$$. 3. **Identify terms:** Let $$a = x^2 - 4y$$ and $$b = x^2 + 4y$$. 4. **Apply the difference of squares formula:** $$ (x^2 - 4y)^2 - (x^2 + 4y)^2 = ((x^2 - 4y) - (x^2 + 4y))((x^2 - 4y) + (x^2 + 4y)) $$ 5. **Simplify each factor:** - First factor: $$ (x^2 - 4y) - (x^2 + 4y) = x^2 - 4y - x^2 - 4y = -8y $$ - Second factor: $$ (x^2 - 4y) + (x^2 + 4y) = x^2 - 4y + x^2 + 4y = 2x^2 $$ 6. **Multiply the simplified factors:** $$ -8y \times 2x^2 = -16x^2 y $$ **Final answer:** $$ -16x^2 y $$