1. **State the problem:** Factor the polynomial $$125j^2 - 5$$ completely. 2. **Identify common factors:** Both terms have a common factor of 5. 3. **Factor out the greatest common factor (GCF):** $$125j^2 - 5 = 5(25j^2 - 1)$$ 4. **Recognize the difference of squares:** $$25j^2 - 1 = (5j)^2 - 1^2$$ 5. **Apply the difference of squares formula:** $$a^2 - b^2 = (a - b)(a + b)$$ 6. **Factor the expression inside the parentheses:** $$25j^2 - 1 = (5j - 1)(5j + 1)$$ 7. **Write the fully factored form:** $$125j^2 - 5 = 5(5j - 1)(5j + 1)$$ **Final answer:** $$5(5j - 1)(5j + 1)$$