1. **State the problem:** Solve the quadratic equation $$2x^2 - 5x - 12 = 0$$ completely. 2. **Formula and rules:** To solve a quadratic equation of the form $$ax^2 + bx + c = 0$$, we can factor it if possible or use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Factor the quadratic:** We look for two numbers that multiply to $$2 \times (-12) = -24$$ and add to $$-5$$. These numbers are $$-8$$ and $$3$$. 4. **Rewrite the middle term:** $$2x^2 - 8x + 3x - 12 = 0$$ 5. **Group terms:** $$(2x^2 - 8x) + (3x - 12) = 0$$ 6. **Factor each group:** $$2x(x - 4) + 3(x - 4) = 0$$ 7. **Factor out the common binomial:** $$(2x + 3)(x - 4) = 0$$ 8. **Set each factor equal to zero and solve:** $$2x + 3 = 0 \Rightarrow x = -\frac{3}{2}$$ $$x - 4 = 0 \Rightarrow x = 4$$ **Final answer:** $$x = -\frac{3}{2}$$ or $$x = 4$$