1. **State the problem:** Factor the quadratic expression $x^2 + 14x + 45$. 2. **Recall the factoring formula:** For a quadratic expression $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add to $b$. 3. **Identify coefficients:** Here, $a=1$, $b=14$, and $c=45$. 4. **Find two numbers:** We need two numbers that multiply to $1 \times 45 = 45$ and add to $14$. 5. **List factor pairs of 45:** $(1, 45), (3, 15), (5, 9)$. 6. **Check sums:** $1+45=46$, $3+15=18$, $5+9=14$. 7. **Select the pair:** $5$ and $9$ satisfy the sum condition. 8. **Rewrite the middle term:** $x^2 + 5x + 9x + 45$. 9. **Group terms:** $(x^2 + 5x) + (9x + 45)$. 10. **Factor each group:** $x(x + 5) + 9(x + 5)$. 11. **Factor out common binomial:** $(x + 5)(x + 9)$. 12. **Final answer:** $$x^2 + 14x + 45 = (x + 5)(x + 9)$$