1. **State the problem:** Solve the quadratic equation $$x^2 + 5x - 3 = 0$$ 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 $a=1$, $b=5$, and $c=-3$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4 \times 1 \times (-3) = 25 + 12 = 37$$ Since $\Delta > 0$, there are two distinct real roots. 4. **Find the roots:** $$x = \frac{-5 \pm \sqrt{37}}{2}$$ 5. **Final answer:** $$x_1 = \frac{-5 + \sqrt{37}}{2}, \quad x_2 = \frac{-5 - \sqrt{37}}{2}$$ These are the two solutions to the quadratic equation.