Just like in the title, I wonder if there is a way to solve $2x(1+\frac{1}{x^3})+7(1+\frac{1}{x})=0$ without any use of solving cubics including the rational root theorem (with these methods I already got $x=-2,-1,-\frac{1}{2}$).

The original equation is supposed to be related to quadratic substitution (the ones where you do $t=x^2$, like $x^4-2x^2+1=0$).

Thanks in advance for any help!