The quadratic expression \( x^2 + 2x + 4 \) cannot be factored into real linear factors because its discriminant is negative. However, it can be factored using complex numbers.

1. Start by checking the discriminant \( \Delta \):
   \[
   \Delta = b^2 - 4ac
   \]
   For the quadratic \( ax^2 + bx + c = 0 \), we have \( a = 1 \), \( b = 2 \), and \( c = 4 \). Substituting these values into the discriminant formula:
   \[
   \Delta = (2)^2 - 4(1)(4) = 4 - 16 = -12
   \]
   Since \( \Delta = -12 \), the roots are complex.

2. The quadratic formula gives the roots:
   \[
   x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-2 \pm \sqrt{-12}}{2(1)} = \frac{-2 \pm 2i\sqrt{3}}{2}
   \]
   Simplifying:
   \[
   x = -1 \pm i\sqrt{3}
   \]

3. Using these roots, the quadratic can be written as:
   \[
   x^2 + 2x + 4 = \left(x - (-1 + i\sqrt{3})\right)\left(x - (-1 - i\sqrt{3})\right)
   \]
   Or more simply:
   \[
   x^2 + 2x + 4 = (x + 1 - i\sqrt{3})(x + 1 + i\sqrt{3})
   \]

This is the factorization in terms of complex numbers.

Would you like further clarification or a detailed breakdown of this process?

Here are some related questions:
1. How can we determine whether a quadratic has real or complex roots?
2. What are the steps to apply the quadratic formula?
3. How does the discriminant help in identifying the nature of the roots?
4. Can all quadratic equations be factored using real numbers?
5. How do we simplify complex expressions like \( \sqrt{-12} \)?

**Tip:** The discriminant \( b^2 - 4ac \) is a quick way to check whether a quadratic equation has real or complex roots.

Explore the factorization of the quadratic equation x² + 2x + 4 using complex numbers. This solution demonstrates key algebraic concepts, including the quadratic formula, discriminant, and complex root calculations. Suitable for students in Grades 9-12.

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