The given polynomial is $x^2 + 4$.

This polynomial can be factored over the complex numbers, using the difference of squares formula. The difference of squares formula is:

$a^2 - b^2 = (a - b)(a + b)$

To apply this, rewrite $4$ as $2^2$, so the expression becomes:

$x^2 + 4 = x^2 - (-4) = x^2 - (2i)^2$

Now, factor using the difference of squares:

$x^2 + 4 = (x - 2i)(x + 2i)$

So, the factorization of $x^2 + 4$ is $(x - 2i)(x + 2i)$.

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. How do you factor polynomials with real coefficients?
2. What is the difference between real and complex roots?
3. How does the difference of squares formula work?
4. What are imaginary numbers, and how are they used in factoring?
5. How do you solve quadratic equations with complex roots?

Tip: When factoring polynomials with no real solutions, consider using complex numbers to complete the factorization.