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A real cubic $p(\lambda )={\lambda }^3+A{\lambda }^2+B\lambda +C$ has roots $-A,\pm i\sqrt{B}$ when $B>0$ and $AB=C$. For $A>0$ this gives a stable transverse third direction. To conclude a generic Hopf bifurcation as a parameter varies, also require a transverse crossing of the imaginary pair and a nonzero first Lyapunov coefficient. The algebraic imaginary-root condition alone is not sufficient.

Take the stipulated polynomial roots $-2,1,5$. Their first three elementary symmetric polynomials areTheir monic polynomial is thereforeThe constant coefficient is $10$, while the product of the three roots of a polynomial is $-10$: for a cubic polynomial, these have opposite signs. Expanding the factors directly verifies the same polynomial identity.

For the nonmonic polynomial $-3x^3+12x^2+21x-30$, the same ratios give sum $-12/(-3)=4$, pairwise-product sum $21/(-3)=-7$, and product $-(-30)/(-3)=-10$. It is $-3$ times the monic polynomial above, so its roots of a polynomial are the same. Checking only the sum and product would leave the middle elementary symmetric polynomial unchecked; all three are needed to recover a monic polynomial of polynomial degree three.