\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
For each of the following polynomials, decide whether it is irreducible in \(\mathbb{R}[X]\):
  1. \(X^2 + 1\),
  2. \(X^2 - 2\),
  3. \(X^2 + X + 1\),
  4. \(X^3 + 1\).

Capture an image of your work. AI teacher feedback takes approximately 10 seconds.