The polynomial x^2 + 1 is irreducible over the real numbers because it has no real roots, as indicated by its positive discriminant. However, it is reducible over the complex numbers, where it can be factored into (x - i)(x + i). Similarly, the polynomial x^2 + x + 1 is also irreducible over the reals, as its discriminant is negative (-3), indicating no real solutions. This discussion highlights the importance of analyzing discriminants to determine the factorability of polynomials in different number systems. Understanding these concepts is crucial for polynomial factorization in algebra.