Question about reduction mod p method for polynomials

Question

I am a undergradute math student and I was traying to prove the polynomial $p(x)= x^4-10x^2+1$ is irreducible in $\mathbb{Q}[x]$. I used reduction mod p and for every prime p I found a factorizarion in $\mathbb{Q}[x]$. I know $p(x)$ is irreducible, but i have a question about this method. If I try with many primes and anyone works, i have to try until I find one that works? or try with another method.Thanks for any comments.