1. **State the problem:** Factor the quadratic expression $x^2 - 7x + 10$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add to $b$. 3. **Apply to our problem:** Here, $a=1$, $b=-7$, and $c=10$. We need two numbers that multiply to $1 \times 10 = 10$ and add to $-7$. 4. **Find the numbers:** The numbers $-5$ and $-2$ satisfy this because $-5 \times -2 = 10$ and $-5 + (-2) = -7$. 5. **Write the factored form:** Using these numbers, the factorization is $(x - 5)(x - 2).$ 6. **Verify:** Expanding $(x - 5)(x - 2)$ gives $x^2 - 2x - 5x + 10 = x^2 - 7x + 10$, confirming the factorization is correct.