1. **State the problem:** Simplify or factor the quadratic expression $12x^2 + 16x + 12$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for factors of $a \times c$ that add up to $b$. 3. **Calculate:** Here, $a=12$, $b=16$, and $c=12$. So, $a \times c = 12 \times 12 = 144$. 4. **Find two numbers that multiply to 144 and add to 16:** These numbers are 8 and 18, but 8 + 18 = 26, not 16. Try 12 and 12: 12 + 12 = 24, no. Try 9 and 16: 9 + 16 = 25, no. Try 6 and 24: 6 + 24 = 30, no. Try 4 and 36: 4 + 36 = 40, no. Try 3 and 48: 3 + 48 = 51, no. Try 2 and 72: 2 + 72 = 74, no. Try 1 and 144: 1 + 144 = 145, no. Since no integer pair sums to 16, try factoring out the greatest common factor first. 5. **Factor out the greatest common factor (GCF):** The GCF of 12, 16, and 12 is 4. $$12x^2 + 16x + 12 = 4(3x^2 + 4x + 3)$$ 6. **Now factor the quadratic inside the parentheses:** For $3x^2 + 4x + 3$, $a=3$, $b=4$, $c=3$, and $a \times c = 9$. 7. **Find two numbers that multiply to 9 and add to 4:** These numbers are 1 and 3. 8. **Rewrite the middle term:** $$3x^2 + 4x + 3 = 3x^2 + 1x + 3x + 3$$ 9. **Group terms:** $$= (3x^2 + 1x) + (3x + 3)$$ 10. **Factor each group:** $$= x(3x + 1) + 3(3x + 1)$$ 11. **Factor out the common binomial:** $$= (x + 3)(3x + 1)$$ 12. **Write the full factorization:** $$12x^2 + 16x + 12 = 4(x + 3)(3x + 1)$$ **Final answer:** $4(x + 3)(3x + 1)$