1. **Problem statement:** Factorise the quadratic expression $x^2 + 5x + 6$. 2. **Formula and rules:** To factorise a quadratic expression of the form $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add to $b$. 3. **Apply to the problem:** Here, $a=1$, $b=5$, and $c=6$. We need two numbers that multiply to $1 \times 6 = 6$ and add to $5$. 4. **Find the numbers:** The numbers $2$ and $3$ satisfy this because $2 \times 3 = 6$ and $2 + 3 = 5$. 5. **Write the factorised form:** Using these numbers, the factorisation is: $$x^2 + 5x + 6 = (x + 2)(x + 3)$$ --- 1. **Problem statement:** Solve the quadratic equation $x^2 + 5x + 6 = 0$. 2. **Use the factorised form:** From the previous step, we have: $$x^2 + 5x + 6 = (x + 2)(x + 3) = 0$$ 3. **Zero product property:** If a product of two factors is zero, then at least one of the factors must be zero: $$x + 2 = 0 \quad \text{or} \quad x + 3 = 0$$ 4. **Solve each equation:** $$x + 2 = 0 \Rightarrow x = -2$$ $$x + 3 = 0 \Rightarrow x = -3$$ 5. **Final solution:** The solutions to the equation are: $$x = -2 \quad \text{or} \quad x = -3$$