Prove that $$2x^6+12x^5+30x^4+60x^3+80x^2+30x+45=0$$ has no real roots.
I think we need to prove this through Eisenstein's Irreducibility criterion, but I am unable to apply it correctly.
Let $a_0, a_1, ... ,a_n$ be integers. Then, '''Eisenstein's Criterion''' states that the polynomial $a_nx^n+a_{n-1}x^{n-1}+ ... + a_1x+a_0$ has an irreducible factor of degree more than $k$ if:
$ 1) p$ is a prime which divides each of $a_0,a_1,a_2,...,a_{k} $
$ 2) a_{k+1}$ is not divisible by $ p$
$ 3) a_0$ is not divisible by $p^2$
How do we use the above here?