I'm trying to see if the following polynomials are irreducible over $\mathbb Q$:
$f(x) = x^4 - x^2 + 2x -1$
$g(x) = x^3 + 7x^2 -8x +1$
$h(x) = x^4 + x^3 + x^2 + x + 1.$
Now, for $h(x)$, I can write $(x+1)$ for $x$ and get : $x^4 + 5x^3 + 10x^2 + 10x +5$, for which I can set $P=5$ and by Eisenstein, this is irreducible over $\mathbb Z$, therefore irreducible over $\mathbb Q$.
For $g(x)$, since it is cubic and primitive, there is either a root or it is irreducible. I get to the stage $g(x)=(x-a)(x^2 + bx +c)$ but then cannot equate the coefficients.
Similar situation for $f(x)$, I get to the stage $f(x)=(x^2 + ax +b)(x^2 +cx + d)$ but get stuck when trying to figure out the coefficients. Am I doing something wrong here? I would really appreciate it if someone could explain this to me please, since I am wondering if I am overlooking a silly mistake. Thanks.