The exercise asks us to prove that $I = \{ f \in \Bbb R[X,Y,Z] \mid f(a,b,c) = 0, ~\forall\,(a,b,c)\in \Bbb S^2 \}$ is a finitely generated ideal of $\Bbb R[X,Y,Z]$.
Well, clearly $I$ is an ideal of $\Bbb R[X,Y,Z]$. Since $\Bbb R$ is Noetherian, by Hilbert's basis theorem, $\Bbb R[X,Y,Z]$ is also Noetherian, and hence $I$ is finitely generated.
I don't know if I am being paranoid (I'll just delete the question if this is the case), but that was all too easy. Can you check if I am missing something obvious here?