Quadratic form for quartics

The quadratic form is

$ f_2(x_1,x_2,x_3)=\sum_{i,j=1,3} A_{ij} x_i x_j$.

It involves 6 independent components of $A_{ij}$ due to symmetries: $(A_{12}+A_{21}) x_1 x_2$. We can therefore impose $A_{ij}=A_{ji}$.

The quartic form is

$f_4(x_1,x_2,x_3)=\sum_{i,j,k,l=1,3} B_{ijkl} x_i x_j x_k x_l$.

$B_{ijkl}$ has 81 components, but due to the symmetries $B_{ijkl}=B_{jikl}=B_{kjil}=B_{ljki}$ it has only 15 independent components.

By counting the number of independent components it is clear that not any quartic form (with 15 independent components) can be written as the product of two quadratic forms (with 6 independent components each).