1. **State the problem:** Factor the quadratic expression $60v^2 + 54v + 162$. 2. **Identify the greatest common factor (GCF):** The coefficients 60, 54, and 162 share a GCF of 6. 3. **Factor out the GCF:** $$60v^2 + 54v + 162 = 6(10v^2 + 9v + 27)$$ 4. **Factor the quadratic inside the parentheses:** We look for two numbers that multiply to $10 \times 27 = 270$ and add to 9. 5. **Check factor pairs of 270:** The pairs are (1,270), (2,135), (3,90), (5,54), (6,45), (9,30), (10,27), (15,18). None of these pairs add to 9. 6. **Conclusion:** Since no integer factor pairs add to 9, the quadratic $10v^2 + 9v + 27$ cannot be factored further over the integers. 7. **Final factored form:** $$6(10v^2 + 9v + 27)$$ This is the simplest factorization using integers.