Find all real values of the parameter a for which the equation $x^4+2ax^3+x^2+2ax+1=0$ has

Question

Find all real values of the parameter a for which the equation $x^4+2ax^3+x^2+2ax+1=0$ has

1) exactly two distinct negative roots

2) at least two distinct negative roots

I tried to factorize it but didn't get any breakthrough.

Answer 1

Hint: Try $x^4+2ax^3+x^2+2ax+1=(x^2+bx+1)(x^2+cx+1)$.

Answer 2

Hint:

This is a reciprocal equation, so set $y=x+\dfrac1x$. Dividing the equation by $x^2$, the equation can be rewritten as $$x^2+2ax+1+\frac{2a}x+\frac1{x^2}=x^2+\frac1{x^2}+2a\Bigl(x+\frac1x\Bigl)+1=y^2+2ay-1=0.$$ Now, as the reduced discriminant is $\Delta'=a^2+1>0$, this equation in $x$ has two roots with opposite signs, whichs are also the signs of $x$ (if $x$ is real): $$y=-a\pm\sqrt{a^2+1\mathstrut}.$$ Can you continue?