The polynomial x² + 1 has no first-degree factors over the real numbers because:

1. A first-degree factor of p(x) corresponds to a real number a with p(a) = 0. 2. The equation x² + 1 = 0 requires x² = −1, which has no real solution. 3. So no real number makes p(a) = 0, and the factor theorem gives no first-degree real factor. 4. Hence x² + 1 cannot be split into a product of two first-degree real polynomials. 5. The other options relate to surface features and not the no-real-root reason. _Source: SCERT Kerala Std X Mathematics Part-2, Chapter 9-11 (pp 227-246, 2019 ed.)._