To solve the quadratic equation $x^2 + 4x + 5 = 0$, we can use the quadratic formula:

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

Here, $a = 1$, $b = 4$, and $c = 5$.

Step 1: Calculate the discriminant

The discriminant is given by $\Delta = b^2 - 4ac$.

$\Delta = 4^2 - 4(1)(5) = 16 - 20 = -4$

Step 2: Analyze the discriminant

Since the discriminant is negative ($\Delta = -4$), the quadratic equation has two complex solutions.

Step 3: Calculate the solutions

The solutions are:

$x = \frac{-4 \pm \sqrt{-4}}{2(1)} = \frac{-4 \pm 2i}{2}$

Simplifying further:

$x = \frac{-4}{2} \pm \frac{2i}{2} = -2 \pm i$

Final Answer:

The solutions are $x = -2 + i$ and $x = -2 - i$.

Would you like more details or have any questions? Here are some related questions:

1. What are the properties of complex numbers?
2. How do we interpret the roots of a quadratic equation geometrically?
3. What happens when the discriminant is zero?
4. Can you solve a quadratic equation using the completing the square method?
5. How can you factorize quadratic equations when the roots are complex?

Tip: When solving quadratic equations, always check the discriminant first to determine the nature of the roots (real or complex).