The generalized Lax conjecture states that the hyperbolicity cone $C_h$ of every hyperbolic polynomial h is a spectrahedron. An equivalent formulation is that there is a hyperbolic polynomial q such that qh is the determinant of a symmetric matrix pencil, definite at some point, where $C_q$ contains $C_h$. We first sketch the proof for the existence of such a q when we restrict to strictly hyperbolic h and omit the second condition. Then we discuss this construction in the case of plane hyperbolic curves in more detail. The latter is part of a joint work with S. Naldi and D. Plaumann.

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