Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be an entire function which is not a constant function.
Let $p(z) = az^3 + bz^2 + cz + d$ be a polynomial such that $p'(z) = k(z-\alpha)(z-\beta)$ for some $\alpha, \beta, k \in \mathbb{C}$ with $\alpha \neq \beta.$
Then $g(z) = p(f(z)) $ is surjective.
$\textbf{Attempt}$ Clearly, $p$ is not a constant polynomial by the condition on $p'$. Therefore, $g$ is a nonconstant entire function.
By Little Picard's Theorem, $g$ is surjective or there exists $z_0$ such that $g(\mathbb{C}) = \mathbb{C}\setminus \{z_0\}.$
Since $f$ is a nonconstant entire fucntion, by the Little Picard's theorem, either $f(\mathbb{C}) = \mathbb{C}$, or there exists $z_1$ such that $f(\mathbb{C}) = \mathbb{C}\setminus \{z_1\}.$
If $f$ is surjective, then $g = p\circ f$ is surjective because $p$ is also surjective. So assume that $f(\mathbb{C}) = \mathbb{C}\setminus \{z_1\}.$
Set $h(z) = f(z) - z_1.$ Then $h$ is an entire function which is non-vanishing on $\mathbb{C}$. So there exists a nonconstant holomorphic function $H(z)$ such that $$f(z)-z_1=h(z) = e^{H(z)}.$$
To show $g$ is surjective, fix $y \in \mathbb{C}.$ Then I want to find $\gamma$ such that $$a(e^{H(\gamma)}+z_1)^3+b(e^{H(\gamma)}+z_1)^2+c(e^{H(\gamma)}+z_1)+d=af^3(\gamma) + bf^2(\gamma) + cf(\gamma) + d = y.$$
Not sure how to proceed from this point. Please gives some help !!