Factor the expression over the complex number x^2+25

How can we tell that the polynomial is irreducible, when we perform square-completion or use the quadratic formula? Let's try square-completion: Not much to complete here, transferring the constant term is all we need to do to see what the trouble is:

We can't take square roots now, since the square of every real number is non-negative!

Here is where the mathematician steps in: She (or he) imagines that there are roots of -1 (not real numbers though) and calls them i and -i. So the defining property of this imagined number i is that

Now the polynomial  has suddenly become reducible, we can write

3 is greater then both of those numbers