let's solve the problem for the general case, i.e. over where is any prime number. a quadratic polynomial is irreducible iff it has no root. this number is obviously where

is the number of those polynomials of degree 2 which are reducible, i.e. they are in the form clearly can be any element of so there are possibilities for

we also have distinct pairs in it is also possible to have which gives us possibilities. thus thus:

if you're looking for the number ofmonicirreducible polynomials of degree 2, then would be the only possibility for and thus in this case