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x^5 + 2x^4 + x^3 - 2x - 1 = 0.

I am aware that fifth order polynomials are generally not analytically soluble. Are there any clever ways to at least approximately solve this equation without resorting to numerical methods or fourth order taylor approximation which does not capture the asymptotic behaviour.

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# Solution to x + 1/(1 + x)^2 = 1/x^2

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