Let $a,b,c,d$ be four prime numbers. We set the polynomial : $$P(X)=\frac{(1-X^{abc})(1-X^{abd})(1-X^{acd})(1-X^{bcd})(1-X^a)(1-X^b)(1-X^c)(1-X^d)}{(1-X)^2(1-X^{ab})(1-X^{ac})(1-X^{ad})(1-X^{bc})(1-X^{bd})(1-X^{cd})}$$ By numerical tests, i see that $P(X)$ always has at least one negative coefficient, how can i prove it?