This question follows this other question.
Let $y$ be a natural number, $x$ a variable and $$ f(x,y):= \frac{x^{2y}-1}{x+1}$$ and $$ g(x,y):= \frac{f(x,y)^{2y+1}-1}{(f(x,y)-1)(xf(x,y)+1)}.$$
For a given $y$, it is not too difficult to see that $g(x,y)$ is a polynomial function of $x$ with integer coefficients.
It seems that $g(x,y)$ is irreducible over $\mathbb{Z}[X]$, if and only if $2y+1$ is prime. How could this be demonstrated?
As an example, we have $$ g(x,4)= \Big(1+x^2-2 x^3+x^4-2 x^5+3 x^6-2 x^7+2 x^8-2 x^9+x^{10}-x^{11}+x^{12}\Big)\Big(1-3 x+15 x^2-45 x^3+108 x^4-216 x^5+388 x^6-645 x^7+1008 x^8-1487 x^9+2076 x^{10}-2760 x^{11}+3511 x^{12}-4293 x^{13}+5052 x^{14}-5726 x^{15}+6252 x^{16}-6585 x^{17}+6699 x^{18}-6585 x^{19}+6252 x^{20}-5726 x^{21}+5055 x^{22}-4299 x^{23}+3521 x^{24}-2772 x^{25}+2091 x^{26}-1505 x^{27}+1029 x^{28}-666 x^{29}+406 x^{30}-231 x^{31}+120 x^{32}-55 x^{33}+21 x^{34}-6 x^{35}+x^{36}\Big)$$
and indeed $2\cdot 4 +1 =9$ is not a prime number.