IS f(x)= x^4 + x^3 -x^2 - 2x -2 irreducable over Q??

Attempt:

Ive shown that this has no linear factors as +-1 +-2 are not roots.

Then I let F(x)=(x^2+ax+b)(x^2+cx+d)

And I end up with

a +c=1

d+ac+b=-1

ad+bc=-2

bd=-2

But i can seam to show why no a b c d exist (or Exist)

Is there an easier approach?

I have an exam in 12 hours so I hope soeone can help!

Thanks in advance!

Niall