1. **Problem:** Factor the trinomial $n^2 - 8n + 15$ completely. 2. **Formula and rules:** For a quadratic trinomial $ax^2 + bx + c$ where $a=1$, factorization looks for two numbers that multiply to $c$ and add to $b$. 3. **Identify values:** Here, $a=1$, $b=-8$, and $c=15$. 4. **Find two numbers:** We need two numbers that multiply to $15$ and add to $-8$. These numbers are $-3$ and $-5$ because $-3 \times -5 = 15$ and $-3 + (-5) = -8$. 5. **Write factors:** The factorization is $$(n - 3)(n - 5)$$. 6. **Verification:** Expanding $$(n - 3)(n - 5) = n^2 - 5n - 3n + 15 = n^2 - 8n + 15$$ which matches the original trinomial. **Final answer:** $$(n - 3)(n - 5)$$