Just some hints. The functions $f^{\,(k)}$ have the same zeros of the polynomials $P_k:=f^{\,(k)}exp(-Q)$, that satisfy $P_0:=P$ and $P_{k+1}=P_k'+P_kQ'$. In particular $P_k$ has degree $deg(P)+kleft(deg(Q)-1right)$, and this is also the total number of zeros of $f^{\,(k)}$. They may be all real: for instance if $Q:=-x^2$ and $P:=1$ one finds the Hermite polynomials, that are orthogonal, hence have all zeros real and simple.
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