Show that there are infinitely many values of α for which

Question

Show that there are infinitely many values of $\alpha$ for which $x^7 +15x^2-30x+\alpha$ is irreducible in $\mathbb Q[x]$.

Answer 1

I'll assume you mean $f(x)=x^3+15x^2-30x+\alpha$. If $\alpha$ is an odd number, then $f$ is irreducible modulo $2$.

---

Added after edit:

Alas $x^7+x^2+1$ is reducible modulo $2$, but cip999's idea of using Eisenstein still works.