Does this quasipolynomial always have infinitely many roots? Consider the quasipolynomial
$$q(\lambda) = \lambda^3 + a\lambda - b\lambda e^{-\lambda h} + c$$
with $a,b,c,h>0$. Is it always true that $q(\lambda)=0$ has infinitely many complex solutions?
If so, how can I prove it?

Yes. The function $f(z)=z^3+az-bze^{-zh} + c$ is entire of order $1$, if it had only finitely many zeroes, then by Hadamard factorization theorem, $f(z)$ would be of form $f(z) = p(z) e^{q(z)}$ with $p,q$ polynomial and $q$ linear. This is obviously impossible.
