yes, it is you can try to prove it as following.
By Rational root theorem it is easily seen that it has no rational root. then try to show that
and then get a contardiction.
Thanks again, i found f(x+1) =x^4 +6x^2+9x-3 and so it's irreducible by Eisenstein's criterion with p=3.
Do you have any idea on how to do the following;
Let F be a field. Show that any element in the field F(x) can be written uniquely as f(x)/g(x) with f(x) and g(x) coprime polynomials and g(x) monic.
The hint suggests looking at the proof of any rational number can be written as m/n with m and n comprime and n positive.
Regarding the first polynomial
Just a quick note, showing a polynomial of degree 2 or 3 has no roots in a particular field is enoughto say it is irreducible over that field. In particular you ruled them all out over the rationals with rational root test, so you are done. No need to worry about another possible factorization.
Oh, don't apologize, to be honest I didn't even notice the degrees didn't add up.
I hope that you mean your contradiction will be that e cannot be in as you have shown there are not roots, and therefore cannot have a linear term, thus is necessarily irreducible if it has no linear term factors because it has degree 3. I just don't want you to multiply anything out and start solving systems of equations or anything, lol. Just want to make sure we are on the same page.