Useful lemma: a and b are relatively prime if and only if (a-b) and b are relatively prime.

Well, (21n+4)/(14n+3) is irreducible if (21 n+4) and (14n+3) are relatively prime.

And this is the case if (7n +1) and (14n+3) are relatively prime.

And this is the case if (7n +1) and (7n+2) are relatively prime.

And this is the case if (7n +1) and 1 are relatively prime, which they obviously are for all natural n.

Therefore, (21n+4)/(14n+3) is irreducible for all natural n