1. **State the problem:** Solve the quadratic equation $$x^2 + 7x + 5 = 0$$ for $x$. 2. **Formula used:** The quadratic formula to solve $ax^2 + bx + c = 0$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=7$, and $c=5$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 7^2 - 4 \times 1 \times 5 = 49 - 20 = 29$$ Since $\Delta > 0$, there are two distinct real roots. 4. **Apply the quadratic formula:** $$x = \frac{-7 \pm \sqrt{29}}{2}$$ 5. **Final answer:** $$x_1 = \frac{-7 + \sqrt{29}}{2}, \quad x_2 = \frac{-7 - \sqrt{29}}{2}$$ These are the two solutions for $x$ in the given quadratic equation.