1. **State the problem:** Solve the quadratic equation by factoring: $$2x^2 + 9x - 1 = 2x - 7$$ 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$2x^2 + 9x - 1 - 2x + 7 = 0$$ Simplify: $$2x^2 + (9x - 2x) + (-1 + 7) = 0$$ $$2x^2 + 7x + 6 = 0$$ 3. **Use the factoring method:** For a quadratic equation $$ax^2 + bx + c = 0$$, find two numbers that multiply to $$a \times c = 2 \times 6 = 12$$ and add to $$b = 7$$. 4. **Find the pair:** The numbers 3 and 4 satisfy this because $$3 \times 4 = 12$$ and $$3 + 4 = 7$$. 5. **Rewrite the middle term:** $$2x^2 + 3x + 4x + 6 = 0$$ 6. **Group terms:** $$(2x^2 + 3x) + (4x + 6) = 0$$ 7. **Factor each group:** $$x(2x + 3) + 2(2x + 3) = 0$$ 8. **Factor out the common binomial:** $$(2x + 3)(x + 2) = 0$$ 9. **Set each factor equal to zero:** $$2x + 3 = 0 \quad \text{or} \quad x + 2 = 0$$ 10. **Solve for x:** $$2x = -3 \Rightarrow x = \frac{-3}{2}$$ $$x = -2$$ **Final answer:** $$x = -\frac{3}{2} \quad \text{or} \quad x = -2$$