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I'm confused. If I do exactly what you've done above:
sage: reset()
sage: import sympy
sage: var("X T p z")
(X, T, p, z)
sage:
sage: eq1 = 80*X*e^(T*p) + X*e^(2*T*p) == 5*e^(T*p)/(e^(T*p) - 1) + e^(T*p)/(e^(T*p) - e^(2*T))
sage: Xi = eq1.solve(X)
sage: X = Xi[0].right()
sage: X = X.simplify_full()
sage: X = sympy.simplify(X)
sage: X = X.subs(e^(p*T), z)
sage: X = SR(X)
sage:
sage: solve(X.denominator(), z)
[z == e^(2*T), z == -80, z == 1]
sage: factor(X.denominator())
(z - 1)*(z + 80)*(z - e^(2*T))
I seem to get the answers you want.
