I am interested in real solutions to homogeneous polynomials in $\mathbb{R}[x_1, ..., x_n]$.

For the case $n = 2$, we can compute the determinant of the symmetric $2 \times 2$ matrix $[a_{ij}]$, $a_{ij} \in \mathbb{R}$. If this determinant is positive, then the corresponding homogeneous polynomial $a_{11}x_1^2 + 2a_{12} x_1 x_2 + a_{22} x_2^2$ factorises into $(c_1 x_1 + c_2 x_2)(d_1 x_1 + d_2 x_2)$, with $c_k, d_k \in \mathbb{R}$.

Can we make a similar statement in $n = 3$ using the hyperdeterminant of the $3 \times 3 \times 3$ tensor corresponding to a homogeneous polynomial of degree 3? Specifically, if this hyperdeterminant is positive, can we say that it factorises into $(c_1 x_1 + c_2 x_2 + c_3 x_3)(d_1 x_1 + d_2 x_2 + d_3 x_3)(e_1 x_1 + e_2 x_2 + e_3 x_3)$ with $c_k, d_k, e_k \in \mathbb{R}$? And if the hyperdeterminant is negative, is there a sense in which the two zeroes of this polynomial are "complex conjugates" like in the single variable cubic case?

I'm just beginning to learn algebraic geometry and have skimmed through Discriminants, Resultants, and Multidimensional Determinants, but haven't found an answer (probably because I don't know enough algebraic geometry!).