1. **State the problem:** Factor the trinomial $$x^2 + 7x + 12$$. 2. **Formula and rules:** To factor a trinomial of the form $$x^2 + bx + c$$, find two numbers that multiply to $$c$$ and add to $$b$$. 3. **Apply to the problem:** Here, $$b = 7$$ and $$c = 12$$. 4. **Find factors of 12 that add to 7:** The pairs are (1,12), (2,6), (3,4). Only (3,4) adds to 7. 5. **Write the factorization:** $$x^2 + 7x + 12 = (x + 3)(x + 4)$$. 6. **Check by expansion:** $$(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$$. **Final answer:** $$(x + 3)(x + 4)$$.