1. **Problem:** Factorise and solve the quadratic equation $2x^2 + 11x + 12 = 0$. 2. **Formula and rules:** To solve quadratic equations by factorisation, we look for two numbers that multiply to $a \times c$ and add to $b$ in $ax^2 + bx + c = 0$. 3. **Step-by-step solution:** - Here, $a=2$, $b=11$, $c=12$. - Calculate $a \times c = 2 \times 12 = 24$. - Find two numbers that multiply to 24 and add to 11: 8 and 3. - Rewrite middle term: $2x^2 + 8x + 3x + 12 = 0$. - Group terms: $(2x^2 + 8x) + (3x + 12) = 0$. - Factor each group: $2x(x + 4) + 3(x + 4) = 0$. - Factor out common binomial: $(2x + 3)(x + 4) = 0$. 4. **Solve each factor:** - $2x + 3 = 0 \Rightarrow 2x = -3 \Rightarrow x = \frac{-3}{2}$. - $x + 4 = 0 \Rightarrow x = -4$. 5. **Intermediate cancellation step:** $$2x + 3 = 0 \Rightarrow \cancel{2}x = \frac{-3}{\cancel{2}}$$ 6. **Final answer:** $$x = -\frac{3}{2} \text{ or } x = -4$$ This completes the factorisation and solution of the first quadratic equation.