1. **State the problem:** Factorize completely the expression $$x^4 - x^2$$. 2. **Identify the common factor:** Notice that both terms have a common factor of $$x^2$$. So, we can factor out $$x^2$$: $$x^4 - x^2 = x^2(x^2 - 1)$$. 3. **Recognize the difference of squares:** The expression inside the parentheses, $$x^2 - 1$$, is a difference of squares since $$1 = 1^2$$. The difference of squares formula is: $$a^2 - b^2 = (a - b)(a + b)$$. 4. **Apply the difference of squares formula:** $$x^2 - 1 = (x - 1)(x + 1)$$. 5. **Write the complete factorization:** $$x^4 - x^2 = x^2(x - 1)(x + 1)$$. 6. **Compare with the options:** - Option A: $$x(x - 1)(x + 1)$$ (missing one $$x$$ factor) - Option B: $$(x^2 - 1)(x^2 + 1)$$ (incorrect factorization) - Option C: $$x^2(x^2 - 1)$$ (partially factored, not fully) - Option D: $$(x^2 - x)(x^2 + x)$$ (incorrect) The fully factorized form matches option C, but since $$x^2 - 1$$ can be further factored, the complete factorization is $$x^2(x - 1)(x + 1)$$ which is not exactly listed. However, option C is the correct intermediate step. **Final answer:** $$x^2(x^2 - 1)$$ (Option C)