1. **Stating the problem:** Solve the quadratic equation $$2x^2 + 5x + 2 = 0$$ using the sum and product method. 2. **Identify coefficients:** For the quadratic equation $$ax^2 + bx + c = 0$$, we have $$a = 2$$, $$b = 5$$, and $$c = 2$$. 3. **Find the product and sum:** - Product = $$a \times c = 2 \times 2 = 4$$ - Sum = $$b = 5$$ 4. **Find two numbers whose product is 4 and sum is 5:** These numbers are $$4$$ and $$1$$ since $$4 \times 1 = 4$$ and $$4 + 1 = 5$$. 5. **Rewrite the middle term using these numbers:** $$2x^2 + 4x + 1x + 2 = 0$$ 6. **Group and factor:** $$ (2x^2 + 4x) + (1x + 2) = 0$$ $$ 2x(x + 2) + 1(x + 2) = 0$$ 7. **Factor out the common binomial:** $$(2x + 1)(x + 2) = 0$$ 8. **Solve for $$x$$:** - Set $$2x + 1 = 0$$ which gives $$x = -\frac{1}{2}$$ - Set $$x + 2 = 0$$ which gives $$x = -2$$ **Final answer:** $$x = -\frac{1}{2}$$ or $$x = -2$$.