To solve the quadratic equation $x^2 + 7x + 3 = 0$, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this equation, $a = 1$, $b = 7$, and $c = 3$. Now, let's plug in these values:

$x = \frac{-7 \pm \sqrt{7^2 - 4(1)(3)}}{2(1)}$

First, calculate the discriminant:

$\Delta = 7^2 - 4(1)(3) = 49 - 12 = 37$

Now substitute this into the formula:

$x = \frac{-7 \pm \sqrt{37}}{2}$

Thus, the two solutions are:

$x_1 = \frac{-7 + \sqrt{37}}{2}, \quad x_2 = \frac{-7 - \sqrt{37}}{2}$

These are the exact solutions. You can leave the answer like this or approximate it:

$x_1 \approx -0.46, \quad x_2 \approx -6.54$

Would you like more details on this solution or further clarification?

Related questions:

1. What does the discriminant tell us about the nature of the roots?
2. How do you complete the square to solve a quadratic equation?
3. Can all quadratic equations be solved using factoring?
4. How would you solve this equation graphically?
5. What are some real-world applications of quadratic equations?

Tip: The discriminant $b^2 - 4ac$ can help you quickly determine whether a quadratic equation has real or complex solutions.