Recurrence sequences as special values of polynomials

Let g(x) be a polynomial with rational coefficients, and F_n the Fibonacci sequence.
When does the equation g(x) = F_n have infinitely many rational solutions x? 
In this lecture, we shall give a full description of such g(x), and further answer a related question regarding the reducibility of polynomials of the form P(F_n, x).
The lecture is based on the speaker's M.Sc. thesis.