Exam 1 Review In Class Expansion

Took a quick look at the problem provided in class of $z+2z^3$ . I think expanding it it is just basic algebra but I was wondering if there was an easier way maybe converting to polar with Euler's to avoid the cubic.
Here's what I tried quickly:
$$\begin{aligned} &z+2z^3\\ &(a+bi)+2(a+bi)^3\\ &(a+bi)+2[(a^3-3ab^2)+i(3a^2b-b^3)]\\ &(a+bi)+2a^3-6ab^2+6a^2bi-2b^3i\\ &(2a^3-6ab^2+a)+i(6a^2b-2b^3+b)\end{aligned}$$

@DPR Looks good to me! I don't think there's any particular advantage to using polar form.