1. **State the problem:** Factor the quadratic expression $2u^2 + 9u + 7$ 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, $a=2$, $b=9$, $c=7$. So, $a \times c = 2 \times 7 = 14$. 4. **Find two numbers that multiply to 14 and add to 9:** These numbers are 7 and 2. 5. **Rewrite the middle term using these numbers:** $$2u^2 + 7u + 2u + 7$$ 6. **Group terms:** $$(2u^2 + 7u) + (2u + 7)$$ 7. **Factor each group:** $$u(2u + 7) + 1(2u + 7)$$ 8. **Factor out the common binomial:** $$(u + 1)(2u + 7)$$ **Final answer:** The completely factored form is $$(u + 1)(2u + 7)$$.