It is easy to construct a polynomial of degree four with integer coefficients that doesn't have any real roots, if it has a real root then it can be factored by division by $x-a$ where $a$ is the root (maybe it's difficult to find the root, if it is not an integer number, but there are a lot of techniques for this).
To construct such a polynomial we can multiply two irreducible (with no real roots also) quadratic polynomials. Then I want to factor this polynomial without knowing any of this two quadratic irreducible polynomials, what is the best approach for solving this?
I constructed this polynomial
Computers (wolfram alpha, etc) can factor this, but how?
And what is the field of algebra that I have to study to answer such questions?