First, we can split the integral into separate integrals for each term:
$$\int\left(2+x+2 x^{2}+e^{x}\right) d x = \int 2 dx + \int x dx + \int 2x^2 dx + \int e^x dx$$
Now, we can find the indefinite integral for each term:
$$\int 2 dx = 2x + C_1$$
$$\int x dx =
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