Given any $\beta \in R$, do there exist integers $k,l,m$, and a real number $x$ such that:
as well as
From the first equation, $x$ is algebraic. This implies $\beta$ algebraic, which might not hold.
There are countably many values of $\beta$ that give solutions. We can enumerate them by considering all triples $(k,l,m)$, finding the roots in $x$ and computing $\beta=x^4-x^2-k$.