I want to show that $f(x) = 7x^4 + 103x^3 + 802x^2 + 6x + 75$ is irreducible in $\mathbb{Q}[x]$. My attempts are:
Consider $f(x)$ reduced mod $2$. This gives $x^4 + x^3 + 1$ which is irreducible as it has no root and is not divisible by $x^2 + x + 1$. So we know that $f(x)$ has no integer roots. Now suppose that $f(x) = g_1(x) g_2(x)$ for $g_1,g_2 \in \mathbb{Z}[x]$. Considering the reduction mod $2$ yields a contradiction so we have that $f(x)$ is irreducible in $\mathbb{Z}[x]$.
I would like to apply now Gauss' Lemma, but the coefficients are not coprime, so I can't. Can someone give an alternative approach?