1. **State the problem:** Factorise fully the expression $$32p^2 x^3 - 2s^2 x^3$$. 2. **Identify common factors:** Both terms have a factor of $$2x^3$$. 3. **Extract the common factor:** $$32p^2 x^3 - 2s^2 x^3 = 2x^3(\cancel{16p^2} - \cancel{1}s^2)$$ 4. **Simplify inside the parentheses:** $$2x^3(16p^2 - s^2)$$ 5. **Recognize difference of squares:** $$16p^2 - s^2 = (4p)^2 - (s)^2$$ 6. **Apply difference of squares formula:** $$a^2 - b^2 = (a - b)(a + b)$$ 7. **Factorize:** $$2x^3(4p - s)(4p + s)$$ **Final answer:** $$\boxed{2x^3(4p - s)(4p + s)}$$