1. **State the problem:** Factor the quadratic expression $3x^2 + 7x + 4$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Calculate the product and sum:** Here, $a=3$, $b=7$, $c=4$. $$a \times c = 3 \times 4 = 12$$ We need two numbers that multiply to 12 and add to 7. 4. **Find the numbers:** The numbers are 3 and 4 because $3 \times 4 = 12$ and $3 + 4 = 7$. 5. **Rewrite the middle term:** $$3x^2 + 3x + 4x + 4$$ 6. **Group terms:** $$(3x^2 + 3x) + (4x + 4)$$ 7. **Factor each group:** $$3x(x + 1) + 4(x + 1)$$ 8. **Factor out the common binomial:** $$(3x + 4)(x + 1)$$ **Final answer:** $$3x^2 + 7x + 4 = (3x + 4)(x + 1)$$