<s>[INST] Given that $x^3+x^2=1$, express the infinite product $(1+x)(1+x^2)(1+x^4)(1+x^8)\ldots$ in the form $A+Bx+Cx^2$. 
Given that $x^3+x^2=1$ and $x\in\mathbb{R}$, express the infinite product $$(1+x)(1+x^2)(1+x^4)(1+x^8)\ldots$$
  in the form $A+Bx+Cx^2$.

In the earlier parts of the question, I have already shown that
$$x^4=-1+x+x^2$$
$$x^{-1}=x+x^2$$
$$1-x+x^2-x^3+x^4-x^5+\ldots=x^2$$
$$\frac{1}{1-x}=2+2x+x^2$$
I also know that
$$1+x=\frac{1}{1-x+x^2-x^3+x^4-x^5+\ldots}.$$
Can anyone give me a hint? Perhaps there is a way to do it using the previous parts, but I cannot see how.
 [/INST] Hint. One may recall that
$$
(1+x)(1+x^2)(1+x^4)(1+x^8)\cdots=\frac1{1-x}, \qquad |x|<1,
$$ then insert
$$
x=\frac1{x+x^2}
$$ in the preceding identity.
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