<s>[INST] How to prove that $X^4+8X^3+X^2+2X+5$ is irreducible? How to prove that $X^4+8X^3+X^2+2X+5$ is irreducible on $\mathbb Q[X]$?
I can try to find $a,b,c,d$ such that
$$X^4+8X^3+X^2+2X+5=(X^2+aX+b)(X^2+cX+d)$$ 
and prove that $a,b,c,d\notin\mathbb Q$ and find $a,b,c,d$ such that
$$X^4+8X^3+X^2+2X+5=(X-a)(X^3+bX^2+cX+d)$$
and prove that $a,b,c,d\notin \mathbb Q$, but it's really long and I would like to know if there is an other method. 
 [/INST] Let $P(X)=A(X)B(X)$ a decomposition of your polynomial into quadratics. Observe that
$$
P(X)\equiv X(X^3+3X^2+X+2) \bmod 5
$$
and the deg 3 factor is irreducible mod 5 because it has no roots mod 5. Thus one of the quadratic factors $A, B$ has only one root mod 5, but this is not possible. Hence there's no factorization.
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