1. **State the problem:** Simplify the expression $$(X^2 + 4)^2 - (X^2 - 2)^2$$. 2. **Formula used:** This is a difference of squares, which follows the rule $$a^2 - b^2 = (a - b)(a + b)$$. 3. **Apply the formula:** Let $$a = X^2 + 4$$ and $$b = X^2 - 2$$, so $$ (X^2 + 4)^2 - (X^2 - 2)^2 = (a - b)(a + b) $$ 4. **Calculate each factor:** $$ a - b = (X^2 + 4) - (X^2 - 2) = X^2 + 4 - X^2 + 2 = 6 $$ $$ a + b = (X^2 + 4) + (X^2 - 2) = X^2 + 4 + X^2 - 2 = 2X^2 + 2 $$ 5. **Multiply the factors:** $$ (a - b)(a + b) = 6(2X^2 + 2) = 12X^2 + 12 $$ 6. **Final answer:** $$ (X^2 + 4)^2 - (X^2 - 2)^2 = 12X^2 + 12 $$ This simplification uses the difference of squares formula to factor and then simplify the expression efficiently.