If polynomials are almost surjective over a field, is the field algebraically closed?

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1. What the problem is asking (in plain language)

*We are given an infinite field (K).
We say that polynomials are almost surjective over (K) if every non‑constant polynomial

[ f(x)\in K[x] ]

has the property that the set of values it misses in (K) is finite; i.e. for all but finitely many (a\in K) the equation

[ f(x)=a ]

has a solution (x\in K).

The question is:

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