How to compute $\min_{\{a, b, c\}}\int^{\infty}_0|x^3-a-bx-cx^2|^2e^{-x}\,dx$ I have no idea with $\min_{\{a, b, c\}}\int^{\infty}_0|x^3-a-bx-cx^2|^2e^{-x}\,dx$, especially the $e^{-x}$ part.

Let $V$ be the inner product space whose elements are polynomial functions on $[0,\infty)$, with inner product defined by $\langle f,g\rangle =\int_0^\infty f(x)\overline{g(x)} e^{-x}dx$.    You are trying to find (the square of) the distance from $x^3$ to the subspace $P_2$ of polynomials of degree at most $2$.  One way to do so is to apply Gram Schmidt on $(1,x,x^2,x^3)$ to express everything in terms of orthonormal vectors.  That way, the integral can be expressed as a sum of squares, where it is easier to see how it is minimized. 
Essentially, this allows you to decompose $x^3$ as $x^3 = f(x) + g(x)$ with $g\in P_2$ and $f\in P_2^\perp$, and then the min in question is $\|f\|^2 =\int_0^\infty |f(x)|^2e^{-x}dx$.
