Finite order function in the complex analysis.

Question

Assume that an entire function $f$ be finite order with finitely many zeros.

Please show that either $f(z)$ is a polynomial or $f(z) + z$ has infinitely many zeros.

Thank you.

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And I know the following theorem,

Suppose f is entire function of finite order. Then either f has infinitely many zeros or $f(z)$ is of the form $Q(z)e^{P(z)}$ for polynomials $P$ and $Q$.

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Answer 1

Finite order is not needed. Apply Picard's "big" theorem to $f(z)/z$.