prime polynomial
Just as a prime number like 7 cannot be split into smaller whole-number factors, a prime polynomial cannot be broken into a product of lower-degree polynomials — within the number system you are allowed to use. It is a dead end for factoring, and recognizing one tells you to stop searching and reach for other tools.
The crucial phrase is 'over the given number system,' because primeness depends on what numbers you permit. For instance, x^2 - 2 is prime over the rationals (no rational factoring exists), but over the reals it factors as (x - sqrt(2))(x + sqrt(2)). Likewise x^2 + 1 is prime over the reals but factors as (x + i)(x - i) over the complex numbers. So 'prime' is always relative to a setting.
In ordinary algebra courses, when we say a quadratic trinomial is prime we usually mean it has no factorization with integer (or rational) coefficients — equivalently, its discriminant is not a perfect square. That is your signal to switch to the quadratic formula or completing the square to find its irrational or complex roots.
x^2 + x + 1 is prime over the rationals: its discriminant is 1 - 4 = -3, not a perfect square, so no rational factoring exists.
Discriminant not a perfect square ⇒ no rational factors.
Calling a polynomial 'prime' is shorthand for 'irreducible over a specified ring or field.' Always know which number system you are working in before you declare a polynomial prime.