1. **State the problem:** Factor the quadratic expression $x^2 - 5x + 6$ fully. 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. Here, $a=1$, $b=-5$, and $c=6$. We need two numbers that multiply to $1 \times 6 = 6$ and add to $-5$. 4. The numbers are $-2$ and $-3$ because $-2 \times -3 = 6$ and $-2 + -3 = -5$. 5. Rewrite the middle term using these numbers: $$x^2 - 2x - 3x + 6$$ 6. Group terms: $$(x^2 - 2x) + (-3x + 6)$$ 7. Factor each group: $$x(x - 2) - 3(x - 2)$$ 8. Factor out the common binomial: $$(x - 3)(x - 2)$$ **Final answer:** $$\boxed{(x - 3)(x - 2)}$$