A real polynomial of degree more than or equal to 3 is reducible, but does it necessarily have a real zero?

Question

Takumi Murayama says "Every polynomial in $\mathbb R[x]$ of degree at least 3 has a real root, and therefore is not irreducible". I think I understand why it is not irreducible, but what's the real root of $f(x)=(x^2+1)(x^2+2)$?

If he is right, then why?

If it is wrong, then what is probably meant? I think this has something to do with complex roots in pairs.

Answer 1

From the comments

The author's claim (on p. 39) is wrong. What he meant is that every polynomial of degree three or more has irreducible factors which are at most quadratic: "So, every maximal ideal is of the form (f) for f a linear polynomial or an irreducible quadratic polynomial." – Michael Hoppe Dec 22 at 12:09