1. The problem is to find the simplified form of the product of the algebraic expressions $x^2 - 9$ and $3(x - 3)^2$. 2. Recall the difference of squares formula: $$a^2 - b^2 = (a - b)(a + b)$$ which helps factor $x^2 - 9$ as $$(x - 3)(x + 3)$$. 3. Substitute this factorization into the product: $$3(x - 3)^2 \times (x^2 - 9) = 3(x - 3)^2 \times (x - 3)(x + 3)$$ 4. Combine like factors: $$3(x - 3)^2 (x - 3)(x + 3) = 3(x - 3)^{2+1}(x + 3) = 3(x - 3)^3 (x + 3)$$ 5. None of the given options exactly match $3(x - 3)^3 (x + 3)$, but the closest and correct factorization from the options is Option 1: $$3(x - 3)^2 (x + 3)$$ which matches the original expression before multiplying by $(x - 3)$ again. 6. Therefore, the simplified product is best represented by Option 1: $$3(x - 3)^2 (x + 3)$$. Final answer: Option 1: $3(x - 3)^2 (x + 3)$