In addition to what's stated in the title, assume that $x$ has a symmetric distribution, so that $E[(x-\mu)^3]=0$.
So the full question is:
How can I find the linear projection, $L(y\mid 1,x)$ when $E(y\mid x) = \alpha + \beta_1 (x-\mu) + \beta_2(x-\mu)^2$ with $\mu = E(x)$, and where $x$ has a symmetric distribution?
I don't really know how to approach this. I believe the first two terms are easy (but perhaps I'm wrong because I'm just using intuition here), $$ L(y\vert 1,x) = (\alpha -\mu) + \beta_1 x $$ because these are linear and in the span of $x$.
I'm not sure how to handle the third term though; I've been trying to use some clever multiplication or iteration to turn it into a $(x-\mu)^3$ term, but that messes up the other two terms. Plus, even if I get it to $(x-\mu)^3$, I don't know where I'd get an unconditional expectation from (I can get a conditional expectation from iteration properties of linear projection and conditional expectation).
In this case I'd prefer a hint over an answer, if possible.
Thanks.