## Factoring (x^2 + 4)

The expression **(x^2 + 4)** cannot be factored using real numbers. Here's why:

### Understanding Factoring

Factoring involves breaking down an expression into its simpler multiplicative components. We typically look for two binomials that multiply together to give the original expression.

### The Problem with (x^2 + 4)

**No Real Roots:**The quadratic equation x^2 + 4 = 0 has no real solutions. This means there are no real numbers that, when squared, result in -4.**Difference of Squares:**Factoring often relies on the difference of squares pattern: (a^2 - b^2) = (a + b)(a - b). However, (x^2 + 4) is a sum of squares, not a difference.

### Factoring with Complex Numbers

While it can't be factored using real numbers, (x^2 + 4) can be factored using **complex numbers**. Complex numbers involve the imaginary unit "i," where i^2 = -1.

**Here's how it works:**

**Rewrite the expression:**x^2 + 4 = x^2 - (-4)**Introduce the imaginary unit:**x^2 - (-4) = x^2 - (2i)^2**Apply the difference of squares pattern:**x^2 - (2i)^2 = (x + 2i)(x - 2i)

Therefore, the factored form of (x^2 + 4) using complex numbers is **(x + 2i)(x - 2i)**.