I assume from the way that the problem is phrased that the scalars here are the real numbers. If so, let (irreducible) and let . Then (reducible).
With a bit more effort, you can get a similar example with rational scalars.
Suppose is an irreducible in R. If is any other polynomial, does that imply that is also irreducible?
I'm particularily interested in the case when .
Here are some examples I've tried
is a constant
So if I choose (c,d) such that the above has integer roots then I've solved the problem.
So I need to be a square. But divides this, so I just need for to be a square.
This is a contradiction to the condition that be irreducible because then g(x) would then be a difference of squares.
I stopped after this......Naturally the next f(x) to try would be a quadratic, but I don't know enough about quartic polynomials to know the conditions for when they're irreducible or not.