1. **Problem:** Factorise the quadratic expression $x^2 + 7x + 12$. 2. **Formula and rules:** To factorise a quadratic expression of the form $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$. 3. **Step-by-step solution:** - We need two numbers that multiply to $12$ and add to $7$. - The pairs of factors of $12$ are $(1, 12), (2, 6), (3, 4)$. - Among these, $3 + 4 = 7$. 4. **Write the factorised form:** $$x^2 + 7x + 12 = (x + 3)(x + 4)$$ 5. **Explanation:** - The factorisation works because when you expand $(x + 3)(x + 4)$, you get $x^2 + 4x + 3x + 12 = x^2 + 7x + 12$. **Final answer:** $$\boxed{(x + 3)(x + 4)}$$