1. **State the problem:** Factor the quadratic expression $$3x^2 + 23x + 14$$ completely. 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, $$a = 3$$, $$b = 23$$, and $$c = 14$$. Calculate $$a \times c = 3 \times 14 = 42$$. We need two numbers that multiply to 42 and add to 23. 4. **Find the pair:** The numbers 21 and 2 satisfy this because $$21 \times 2 = 42$$ and $$21 + 2 = 23$$. 5. **Rewrite the middle term:** $$3x^2 + 21x + 2x + 14$$ 6. **Group terms:** $$(3x^2 + 21x) + (2x + 14)$$ 7. **Factor each group:** $$3x(x + 7) + 2(x + 7)$$ 8. **Factor out the common binomial:** $$(3x + 2)(x + 7)$$ 9. **Final answer:** $$3x^2 + 23x + 14 = (3x + 2)(x + 7)$$