1. **State the problem:** Factor the expression $24n^3 + 68n^2 + 48n$. 2. **Identify the common factors:** Each term contains a factor of $n$, so factor out $n$ first. 3. **Factor out the greatest common factor (GCF):** $$24n^3 + 68n^2 + 48n = n(24n^2 + 68n + 48)$$ 4. **Factor the quadratic inside the parentheses:** We want to factor $24n^2 + 68n + 48$. 5. **Find the GCF of the quadratic coefficients:** 24, 68, and 48 are all divisible by 4. 6. **Factor out 4:** $$n(24n^2 + 68n + 48) = n \times 4 (6n^2 + 17n + 12) = 4n(6n^2 + 17n + 12)$$ 7. **Factor the quadratic $6n^2 + 17n + 12$:** We look for two numbers that multiply to $6 \times 12 = 72$ and add to 17. These numbers are 9 and 8. 8. **Rewrite the middle term:** $$6n^2 + 9n + 8n + 12$$ 9. **Group terms:** $$(6n^2 + 9n) + (8n + 12)$$ 10. **Factor each group:** $$3n(2n + 3) + 4(2n + 3)$$ 11. **Factor out the common binomial:** $$(3n + 4)(2n + 3)$$ 12. **Write the fully factored form:** $$4n(3n + 4)(2n + 3)$$ **Final answer:** $4n(3n + 4)(2n + 3)$