BALTIC-WAY 2018 Problem 5

A polynomial $f(x)$ with real coefficients is called generating, if for each polynomial $\varphi(x)$ with real coefficients there exist a positive integer $k$ and polynomials $g_{1}(x), \ldots, g_{k}(x)$ with real coefficients such that

$$\varphi(x)=f\left(g_{1}(x)\right)+\cdots+f\left(g_{k}(x)\right) .$$

Find all generating polynomials.