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. 

A prime number is one where its only factors are itself and one. The factoring you performed gives two factors, so to be prime one of these must be 1 or -1.
Solving $(2x+1)=1$ gives us $x=0$ and the other factor is $-1$. This isn't prime.
Solving $(2x-1)=1$ gives us $x=1$ and the other factor is $3$. This is the only prime of that form.
Solving $(2x+1)=-1$ gives us $x=-1$ and the other factor is $3$, which works.
Solving $(2x-1)=-1$ gives us $x=0$ again.
So the candidates are $x=1,-1$.
