Find all values of a for which the equation $x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0$ possesses at least two distinct negative roots Find all values of a for which the equation $$x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0 $$ possesses at least two distinct negative roots.
I am able to prove that all roots would be negative .How to proceed after this.

This is a symmetric (the coefficients) polynomial. We may want to take advantage of that. We can divide by $x^2$ and get $$(x+\frac{1}{x})^2+(a-1)(x+\frac{1}{x})-1=0$$
From there we can solve for $x+\frac{1}{x}$ and get $$x+\frac{1}{x}=\frac{-(a-1)\pm\sqrt{(a-1)^2+4}}{2}.$$
This is a quadratic equation. Solve for $x$ and get the roots.
