1. **State the problem:** Solve the quadratic equation $$2x^2 + 7x + 3 = 0$$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a=2$, $b=7$, and $c=3$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 7^2 - 4 \times 2 \times 3 = 49 - 24 = 25$$. 5. **Apply the quadratic formula:** $$x = \frac{-7 \pm \sqrt{25}}{2 \times 2} = \frac{-7 \pm 5}{4}$$. 6. **Find the two solutions:** - For the plus sign: $$x = \frac{-7 + 5}{4} = \frac{-2}{4} = -\frac{1}{2}$$. - For the minus sign: $$x = \frac{-7 - 5}{4} = \frac{-12}{4} = -3$$. 7. **Final answer:** The solutions are $$x = -\frac{1}{2}$$ and $$x = -3$$.