Irreducibility in $\mathbb{R}[X]$ and $\mathbb{C}[X]$

Over $\mathbb{C}$. Every non-constant polynomial has a root in $\mathbb{C}$ (fundamental theorem of algebra). So the only irreducible polynomials are the linear ones: $aX+b$ with $a\neq0$.

Over $\mathbb{R}$. Irreducible polynomials are:
- Linear polynomials $aX+b$ ($a\neq0$);
- Quadratic polynomials $aX^2+bX+c$ with $a\neq0$ and $\Delta=b^2-4ac<0$.

Why? Any real polynomial factors into linear and quadratic factors by pairing conjugate complex roots.

Conclusion. Over $\mathbb{C}$: linear only. Over $\mathbb{R}$: linear and quadratic with negative discriminant.