proving that $8x^3-6x-1$ is irreducible over $\mathbb{Q}$

Question

When considering the impossibility of trisecting the 60 degree angle, one comes across the polynomial $f(x)=8x^3-6x-1$, which I want to prove is irreducible over $\mathbb{Q}$. I reduced the coefficients mod 5 and obtained $3x^3+4x+4$, which happens to be irreducible over $\mathbb{Z}_5$ by brute force calculation: none of the 5 elements of $\mathbb{Z}_5$ are zeros, and since the polynomial has degree 3 and no linear factors, it is irreducible over $\mathbb{Z}_5$. By the mod $p$ irreduciblity test, I concluded that $f(x)$ is irreducible over $\mathbb{Q}$. Is my proof correct? Some of the solutions online use the rational roots theorem instead.