Let R be any ring.
An ideal I is maximal iff R/I is a field.
An ideal I is prime iff R/I is an integral domain
If you accept the theorem above (particularly the first one), all you'll need to show is that the irreducible polynomials are precisely the prime elements of F[x]. If you haven't read about the theorem above, or you're having trouble showing irreducible polynomials are prime, I can sketch the proof.