Polynomial factorization into irreducibles over $\mathbb{Q}[x]$ I need to find irreducible factors of $f(x)=x^4+3x^3+2x^2+1$ in $\mathbb{Q}[x]$ and explicitely prove that these factors are indeed irreducible.
I believe we can't reduce $f(x)$ any further but I have to prove that this is the case.
I have already shown that there can't be any linear factors if there was such a factorization using the rational root theorem. But another possibilty is a factorization into two polynomials of degree $2$. So something like: $f(x)=(ax^2+bx+c)(mx^2+nx+p)$. I tried writing it out but since we're working with rationals I find it difficult to find or rule out such polynomials.
What do I have to do in this case? And more general: is there a way to find such second degree factors more easily in $\mathbb{Q}$?

$f$ has degree $4$ and assumes prime values for these $9 > 2 \cdot 4$ points and so must be irreducible: 
$$
\begin{array}{rl}
n & f(n) \\
-8 & 2689 \\
-7 & 1471 \\
-4 & 97 \\
-3 & 19 \\
1 & 7 \\
3 & 181 \\
5 & 1051 \\
6 & 2017 \\
7 & 3529 \\
\end{array}
$$
