1. **State the problem:** Factor the quadratic expression $$147m^2 - 168m + 48$$ completely. 2. **Identify the greatest common factor (GCF):** The coefficients are 147, -168, and 48. The GCF of these numbers is 3. 3. **Factor out the GCF:** $$147m^2 - 168m + 48 = 3(49m^2 - 56m + 16)$$ 4. **Factor the quadratic inside the parentheses:** We want to factor $$49m^2 - 56m + 16$$. 5. **Use the method of factoring trinomials:** Find two numbers that multiply to $$49 \times 16 = 784$$ and add to $$-56$$. 6. The pair is $$-28$$ and $$-28$$ because $$-28 \times -28 = 784$$ and $$-28 + -28 = -56$$. 7. Rewrite the middle term: $$49m^2 - 28m - 28m + 16$$ 8. Group terms: $$(49m^2 - 28m) - (28m - 16)$$ 9. Factor each group: $$7m(7m - 4) - 4(7m - 4)$$ 10. Factor out the common binomial: $$(7m - 4)(7m - 4) = (7m - 4)^2$$ 11. **Write the complete factorization:** $$3(7m - 4)^2$$ **Final answer:** $$\boxed{3(7m - 4)^2}$$