I have two questions. I want to prove $x^3-6x^2+12x+7$ is irreducible in rational numbers. My attempt is to use Gauss's lemma, that a primitive polynomial is irreducible in integers iff it is irreducible in rationals. And in integers, it is enough to prove there's no proper factorization of the polynomial. Next, I assume there is a linear factor, $(x-a)$. Then $a$ must divide $7$. But such an integer doesn't exist. Therefore, the polynomial has no linear factor, thus no quadratic factor. This implies $x^3-6x^2+12x+7$ has no proper factorization in integers, so it is irreducible in rationals.
Please tell me whether my answer is correct.
And for polynomial $x^4+8x^3+24x^2+35x-8$, I used a similar approach. First prove there's no linear factor, and then consider two quadratic factors. By equating the coefficients, I showed that there's no such integers. Then the polynomial has no proper factorization so irreducible.
I want to know does this approach generally work, and is there a technique to use Eisenstein's Criterion to show these two polynomials are irreducible.