Real roots of a quintic polynomial

Consider a real quintic polynomial

$$p(x;\, \alpha, \beta)=a_0 (\alpha,\beta) + a_1 (\alpha,\beta) x + a_2 (\alpha,\beta) x^2+ a_3 (\alpha,\beta) x^3 + a_4 (\alpha,\beta) x^4 - x^5$$
with real valued functions $a_i$ defined by
$$\forall i \in \{1,\ldots, 5\}\quad a_i:\Omega \to \mathbb{R},$$
where $\Omega \subset \mathbb{R}^2$.

I'd like to proof, that $p$ has only real roots in $x$ for all $(\alpha,\beta) \in \Omega$. A proof relying on Sturm's Theorem seems not feasible as the given functions $\alpha_i$ are quite complex expressions themselves. Is there an easier method to accomplish this?

Answer

I assume all $a_i$ are continuous.
Compute the discriminant $D(\alpha,\beta)$ of the polynomial. If the set $D^{-1}(0)\subseteq \Omega$ has no interior points, it is suficient to check a single $(\alpha,\beta)$ per connected component of $\Omega\setminus D^{-1}(0)$.