1. **State the problem:** Solve the quadratic equation $$2x^2 + 7x + 3 = 0$$ by factorization. 2. **Recall the factorization method:** For a quadratic equation $$ax^2 + bx + c = 0$$, we look for two numbers that multiply to $$a \times c$$ and add to $$b$$. 3. **Calculate product and sum:** Here, $$a = 2$$, $$b = 7$$, and $$c = 3$$. Calculate product: $$a \times c = 2 \times 3 = 6$$. We need two numbers that multiply to 6 and add to 7. These numbers are 6 and 1. 4. **Rewrite the middle term:** Split the middle term using 6 and 1: $$2x^2 + 6x + 1x + 3 = 0$$ 5. **Group terms:** $$(2x^2 + 6x) + (1x + 3) = 0$$ 6. **Factor each group:** $$2x(x + 3) + 1(x + 3) = 0$$ 7. **Factor out the common binomial:** $$(2x + 1)(x + 3) = 0$$ 8. **Apply zero product property:** Set each factor equal to zero: $$2x + 1 = 0$$ or $$x + 3 = 0$$ 9. **Solve each equation:** For $$2x + 1 = 0$$: $$2x = -1$$ $$x = \frac{\cancel{2}x}{\cancel{2}} = \frac{-1}{2}$$ For $$x + 3 = 0$$: $$x = -3$$ **Final answer:** $$x = -\frac{1}{2}$$ or $$x = -3$$