1. **Problem statement:** Factorise the quadratic expression $x^2 - 12x + 20$. 2. **Formula and rules:** To factorise a quadratic of the form $x^2 - bx + c$, we look for two numbers that multiply to $c$ and add to $-b$. 3. **Identify values:** Here, $b = 12$ and $c = 20$. 4. **Find two numbers:** We need two numbers whose product is $20$ and sum is $12$. The pairs of factors of $20$ are $(1, 20)$, $(2, 10)$, $(4, 5)$. None of these pairs add to $12$. 5. **Check signs:** Since the middle term is $-12x$, the two numbers must add to $-12$ and multiply to $20$. So we look for two numbers that multiply to $20$ and add to $-12$. These are $-10$ and $-2$ because $-10 \times -2 = 20$ and $-10 + (-2) = -12$. 6. **Write factorised form:** Using these numbers, the factorised form is: $$x^2 - 12x + 20 = (x - 10)(x - 2)$$ 7. **Verification:** Expanding $(x - 10)(x - 2)$ gives: $$x^2 - 2x - 10x + 20 = x^2 - 12x + 20$$ which matches the original expression. **Final answer:** $$(x - 10)(x - 2)$$