Two different roots for $P(x) = x^4+ax^2+bx+c$ 
Let $a, b, c \in \mathbb{R}$ and $a > 0$. Also let $P: \mathbb{R} \to \mathbb{R}$, $P(x) = x^4+ax^2+bx+c$. Show that the function has at most two different roots.

My assumption was to use Bolzano's theorem, but I couldn't figure out how to use it here. Also I'm curious if we could use something like Vieta's formula here? Any help would be appreciated.

We have
$$P'(x)=4x^3+2ax+b$$
and $$P''(x)=12x^2+2a>0$$
this means $P$ is a convex function over $\mathbb{R}$, and has at most two roots.
