1. Problem: Simplify the expression $$2(a^4 + 6)^2$$ given that $$a < 6$$ and the expression $$\sqrt{a - 6}$$ is mentioned but not directly related to simplification. 2. Formula and rules: To simplify expressions like $$2(a^4 + 6)^2$$, we use the rule of exponents and distributive property. The square of a binomial is expanded as $$(x + y)^2 = x^2 + 2xy + y^2$$. 3. Apply the formula: $$2(a^4 + 6)^2 = 2[(a^4)^2 + 2 \cdot a^4 \cdot 6 + 6^2]$$ 4. Calculate each term: $$= 2[a^8 + 12a^4 + 36]$$ 5. Distribute the 2: $$= 2a^8 + 24a^4 + 72$$ 6. Final simplified expression: $$2a^8 + 24a^4 + 72$$ Note: The condition $$a < 6$$ and the expression $$\sqrt{a - 6}$$ do not affect the simplification of this polynomial expression.