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).

Learn how to solve the quadratic equation x^2 + 4x + 5 = 0 with a detailed step-by-step solution. This problem involves complex numbers and covers the quadratic formula for finding roots of quadratic equations. Suitable for students in Grades 10-12.

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