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p_star_1 znd the expression you seek are not equal :

sage: SEx=c/2 + (f^2  - f - 2)/(2*(f^2  - 2*f - 1 ))
sage: (p_star_1-SEx).expand().is_zero()
False

which is more obvious if you ask a typeset form :

sage: view((p_star_1/SEx).expand().factor())

which displays :

$$ \frac{{\left(c f^{2} - 2 \, c f + f^{2} - c - 2 \, f\right)} {\left(f^{2} - 2 \, f - 1\right)}^{2}}{c f^{2} - 2 \, c f + f^{2} - c - f - 2} $$

The problem might be

sage: p_star_1.operands()
[(c + 1)*f^2 - (2*c + 1)*f - c - f, f^2 - 2*f - 1, 1/2]
sage: p_star_1.operands()[0].collect(f)
(c + 1)*f^2 - 2*(c + 1)*f - c

p_star_1 znd the expression you seek are not equal :

sage: SEx=c/2 + (f^2  - f - 2)/(2*(f^2  - 2*f - 1 ))
sage: (p_star_1-SEx).expand().is_zero()
False

which is more obvious if you ask a typeset form :

sage: view((p_star_1/SEx).expand().factor())

which displays :

$$ \frac{{\left(c f^{2} - 2 \, c f + f^{2} - c - 2 \, f\right)} {\left(f^{2} - 2 \, f - 1\right)}^{2}}{c f^{2} - 2 \, c f + f^{2} - c - f - 2} $$

The problem might be

sage: p_star_1.operands()
[(c + 1)*f^2 - (2*c + 1)*f - c - f, f^2 - 2*f - 1, 1/2]
sage: p_star_1.operands()[0].collect(f)
(c + 1)*f^2 - 2*(c + 1)*f - c