TITLE: irreducible polynomials
QUESTION [5 upvotes]: Please help me to prove that these polynomials are irreducible over $\mathbb{Q}$:
1) $x^4+5x^3+7x^2-6$
2) $x^4-4x^3+8x^2-8x+9$
Unfortunately , it takes infinite amount of time to prove that given polynomials are irreducible according to Eisenstein's criterion. So could you recommend and show other ways, that requires less time , because such exercises will come in final exam where I should solve a problems in a short time.

REPLY [6 votes]: Actually Eisenstein is very quick and easy here since the required shift is obvious as I explain below. The remark shows how this is a special case of general method to discover such Eisenstein shifts.
Hint $ $ for $2)\:\, \bmod\color{#c00}2\!:\,\ f(x) \equiv x^{ 4}\!+\!1\equiv (x\!+\!1)^{4} \ $ is a prime power. 
So Eisenstein works on $\,g(x) = f(x\!-\!1)  \equiv x^4\ $ by 
 $\,g(0) = f(-1)\equiv 1\!+\!9\equiv 2\pmod{\!4}$ 
Remark $\ $ Recall that the key idea behind the Eisenstein criteria is that polynomials satisfying the criterion are, mod $\,p,\,$ powers of a prime, viz. $\,\equiv x^n,\,$ and products of primes always factor uniquely. The same works for its  shift $\,(x-c)^n,\,$ so we seek primes $\,p\,$ such that, mod $\,p,\,$ the polynomial is congruent to such a power (e.g. for motivation: cyclotomic case). The only primes $\,p\,$ that can yield such powers are those dividing the discriminant (here by Alpha = $\,\color{#c00}2^9\cdot 3\cdot 5^2)$.  Indeed, if $\,f\equiv a (x-c)^n,\,\ n> 1\,$ then $\,f\,$ and $\,f'\,$ have a common root $\,x\equiv c,\,$  hence their resultant $\, R(f,f')\equiv 0.\,$ But this is, up to sign, the discriminant of $\,f\,$ (presumed monic).