For a homogeneous polynomial with real coefficients:$f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz$, suppose we know $f$ factors into products of linear forms$f=(p\_1x+p\_2y+p\_3z)(q\_1x+q\_2y+q\_3z)$, are there anyway to determinate whether the coefficients of the linear forms are real or complex?



quadratic form factorization
The coefficients are real if and only if (depending on whether or not $p\_1x+p\_2y+p\_3z$ and $q\_1x+q\_2y+q\_3z$ are proportional or not) such a quadratic form can be, by a coordinate change, brought to $\pm x^2$ or to $xy$ (this is an obvious re-phrasing, take the factor(s) for new coordinate(s)). The second one can be brought to $x^2-y^2$ also. In other words, the signature of your forms(number of pluses/minuses when brought to sum of squares) should be (1,0) or (0,1) or (1,1) in order for a factorisation over reals to exist. 


The ways to bring a quadratic form to sum of squares over reals are explained in most linear algebra textbooks. In fact, to compute the signature, you don't even need that. Form the matrix $$\begin{pmatrix}a&\frac12d&\frac12e\\ \frac12d&b&\frac12f\\ \frac12e&\frac12f&c\end{pmatrix}.$$ You want the characteristic polynomial to have two zero roots, or one zero root, one positive root, and one negative root.


