How often must an irreducible polynomial take a prime value?
Solution 1:
The values of the polynomial
$$x^2 + x + 4 = \left(x + \frac{1}{2} \right)^2 + \frac{15}{4}$$
are always divisible by $2$ and always greater than $2$, and thus are never prime.
Solution 2:
This is not known.
Dirichlet proved it for the linear polynomials.
There is a related conjecture: http://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H