Chapter 1, Exercise 1.12: Algebraic geometry breaks down over a non-algebraically closed field

Give an example of an irreducible polynomial $f\in\mathbb R[x,y]$, whose zero set $Z(f)$ in $\mathbb A^2_{\mathbb R}$ is not irreducible (cf. 1.4.2).

This is as easy and pathological as it sounds - take $f(x,y)=x^2+y^2+1$, then $Z(f)=\emptyset$ in $\mathbb A^2_{\mathbb R}$ which is vacuously not irreducible.