1. **State the problem:** Factor the expression $$9a^2x^3 - 16x^3$$. 2. **Identify common factors:** Both terms have a common factor of $$x^3$$. 3. **Factor out the common factor:** $$9a^2x^3 - 16x^3 = x^3(9a^2 - 16)$$ 4. **Recognize the difference of squares:** The expression inside the parentheses is $$9a^2 - 16$$, which is a difference of squares since $$9a^2 = (3a)^2$$ and $$16 = 4^2$$. 5. **Apply the difference of squares formula:** $$A^2 - B^2 = (A - B)(A + B)$$ 6. **Factor the difference of squares:** $$9a^2 - 16 = (3a - 4)(3a + 4)$$ 7. **Write the fully factored form:** $$9a^2x^3 - 16x^3 = x^3(3a - 4)(3a + 4)$$ **Final answer:** $$\boxed{x^3(3a - 4)(3a + 4)}$$