Prove that $x^4 + ax^3 + (b - 2)x^2 - ax + 1$ has $4$ real roots

Question

The quadratic polynomial $x^2 + ax + b$ has exactly $2$ real roots. Prove that the quartic polynomial $x^4 + ax^3 + (b - 2)x^2 - ax + 1$ has $4$ real roots.

I've tried Viete formulas and calculating discriminants, but with no success.

Answer 1

Hint:

Divide the given equation with $x^2$ (obviously you can do that since $0$ is not a solution to this equation).

Write $t =x-{1\over x}$ then $t^2+2 = x^2+{1\over x^2}$ and now we have

$$ t^2+at+b=0$$

Since this one has exactly $2$ real solution ...