How to show that a polynomial is irreducible

Question

1. How do I show that a polynomial is irreducibel?
2. How do I show that $x^2+1$ is irreducible over the field $F_p$ where $p \equiv 3 \mod 4$?

My guess for number 1) is that inserting all numbers $x$ from zero to $p-1$ into the formula, the polynomial is irreducible as long as $f(x)$ is never zero. That somehow only works for irreducible polynomials of degree 2 or 3.

No idea how to solve exercise 2)