1. **State the problem:** Simplify the expression $$(a^2 + 2)^2(-2 - a^2)(2 - a^2)$$. 2. **Recall the formula and rules:** We will use the difference of squares formula: $$ (x - y)(x + y) = x^2 - y^2 $$. 3. **Simplify the product of the last two factors:** $$(-2 - a^2)(2 - a^2) = (-(a^2 + 2))(2 - a^2) = -(a^2 + 2)(2 - a^2)$$ 4. Apply difference of squares to $(a^2 + 2)(2 - a^2)$: $$ (a^2 + 2)(2 - a^2) = (2 + a^2)(2 - a^2) = 2^2 - (a^2)^2 = 4 - a^4 $$ 5. Substitute back: $$ -(a^2 + 2)(2 - a^2) = -(4 - a^4) = -4 + a^4 = a^4 - 4 $$ 6. Now the original expression becomes: $$ (a^2 + 2)^2 (a^4 - 4) $$ 7. Factor $a^4 - 4$ as a difference of squares: $$ a^4 - 4 = (a^2)^2 - 2^2 = (a^2 - 2)(a^2 + 2) $$ 8. Substitute this back: $$ (a^2 + 2)^2 (a^2 - 2)(a^2 + 2) = (a^2 + 2)^3 (a^2 - 2) $$ **Final answer:** $$ (a^2 + 2)^3 (a^2 - 2) $$