1. **State the problem:** Factor the expression $5x^3 - 10$. 2. **Identify the common factor:** Both terms have a common factor of 5. 3. **Apply the distributive property:** Factor out 5 from each term: $$5x^3 - 10 = 5(x^3 - 2)$$ 4. **Check for further factorization:** The expression inside the parentheses, $x^3 - 2$, is a difference of cubes since $2 = \sqrt[3]{2}^3$. 5. **Use the difference of cubes formula:** $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ Here, $a = x$ and $b = \sqrt[3]{2}$. 6. **Write the fully factored form:** $$5(x - \sqrt[3]{2})(x^2 + x\sqrt[3]{2} + (\sqrt[3]{2})^2)$$ 7. **Simplify the last term:** $$(\sqrt[3]{2})^2 = \sqrt[3]{4}$$ 8. **Final answer:** $$5(x - \sqrt[3]{2})(x^2 + x\sqrt[3]{2} + \sqrt[3]{4})$$