1. **State the problem:** Solve the quadratic equation $$2x^2 + x - 6 = 0$$ using factorisation. 2. **Rewrite the equation:** The equation is already in standard form $$ax^2 + bx + c = 0$$ where $$a=2$$, $$b=1$$, and $$c=-6$$. 3. **Factor out the common factor if possible:** Here, no common factor for all terms except 1, so proceed to factorise the quadratic. 4. **Multiply $$a$$ and $$c$$:** $$2 \times (-6) = -12$$. 5. **Find two numbers that multiply to $$-12$$ and add to $$b=1$$:** These numbers are $$4$$ and $$-3$$ because $$4 \times (-3) = -12$$ and $$4 + (-3) = 1$$. 6. **Rewrite the middle term using these numbers:** $$2x^2 + 4x - 3x - 6 = 0$$ 7. **Group terms:** $$(2x^2 + 4x) + (-3x - 6) = 0$$ 8. **Factor each group:** $$2x(x + 2) - 3(x + 2) = 0$$ 9. **Factor out the common binomial:** $$(2x - 3)(x + 2) = 0$$ 10. **Set each factor equal to zero and solve for $$x$$:** $$2x - 3 = 0 \Rightarrow 2x = 3 \Rightarrow x = \frac{3}{2}$$ $$x + 2 = 0 \Rightarrow x = -2$$ **Final answer:** $$x = \frac{3}{2}$$ or $$x = -2$$