Proof that any every operator on a finite-dimensional,nonzero,real vector space has an invariant subspace of dimension 1 or 2

Question

in Steven Roman's Advanced linear algebra, the author prove the theorem in the following way: suppose that f is a real linear operator and then factors its minimal polynomial m(x) into a product of linear and quadratic factors over R, if there is a linear factor $x-\lambda $then we can take the subspace spanned by the eigen vector, otherwise, let p(x)=$x^2+ax+b$ be an irreducible quadratic factor of m(x) and write m(x)=p(x)q(x), since $q(f)\neq0$, we may choose a nonzero vector v such that $q(f)v\neq0$, let $W=span(q(f)v,fq(f)v)$, then I have proved that W is invariant under f, but how to show that W has dimension 2?