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

1)

is a constant

2)

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.