Note: the notation "p_k" means p with a subscript k.
The proof of this problem can be seen as a kind of generalization of Euclid's proof that there are infinitely many primes. You have most likely seen Euclid's proof if you are in a basic undergraduate number theory course.
The solution of this problem, like Euclid's, is a proof by contradiction:
Assume there are only finitely many primes of the form 4n+3.
This is a proof regarding infinite numbers of primes.