Finding the constant coefficients in a simple linear model

Let X~exp(1), Y=e^{-X} and consider the simple linear model $\displaystyle Y= \alpha + \beta X + \gamma X^2 + W$ with $\displaystyle E(W) = 0 = \rho(X, W) = \rho(X^2, W) $

I need to evaluate the constants alpha, beta, and gamma using the information given above. I'm wondering if I am doing this correctly. I showed that Y~unif(0,1) which made calculations a bit easier, then I found E(Y), Cov(X,Y), Cov(X^{2},Y), got 3 equations and solved for the 3 unknowns.

pic of my work:http://i.imgur.com/HHlMl.jpg

moment calculations:http://i.imgur.com/O0m0h.jpg

I can't figure out where I went wrong; the next part of the question involves calculating sigma(W)/sigma(Y), and using my calculated coefficients I am getting a negative number for sigma(W).

Can someone point out where I'm making a mistake and/or how to do part c?