find all integers x such that $4x^2 - 1$ is prime I tried factoring it and got $(2x+1)(2x-1)$, however I do not know how to prove for all integers from here. 

Well, you're trying to show that $4x^2-1$ is prime for some integer $x$, but you just factored it! If $4x^2-1$ is going to be prime, your factorization has to a trivial factorization. We can always factor primes as $p=p\cdot 1$, so maybe this will lead you to the correct answer.
