Proving a quantified statement with $\exists\forall$

Condition. $x^2+ax+a>0$ for all $x$ iff $\Delta=a^2-4a<0$, i.e. $a(a-4)<0$, i.e. $0<a<4$.

Existential. The set of such $a$ is nonempty (e.g. $a=1$), so $\exists a,\ \forall x,\ x^2+ax+a>0$ is true.

Conclusion. The statement holds; the qualifying $a$ are $a\in(0,4)$.