abstract algebra - Show that either $f$ is irreducible or $f$ factors into irreducible polynomials of degree $3$ over $K$.

Let $f(x)$ be an irreducible polynomial of degree $6$ over field $K$.

If $L$ is a field extension of $K$ and $[L: K]=2$ then show that either $f$ is irreducible or $f$ factors into irreducible polynomials of degree $3$ over $K$.

Attempt:

If $f$ is irreducible then we are done.

Otherwise $f$ factors into either $1+5$ or $2+4$ or $3+3$ degree polynomials.

I am unable to derive a contradiction for the first two cases.

Please give some hints.