  By defining multiplicities for zeros of polynomials over hyperfields, Baker
and Lorscheid were able to provide a unifying perspective on Descartes's rule
and the Newton polygon rule for polynomials over a formally-real and valued
field respectively. In this paper, we apply their multiplicity formula to the
hyperfield associated with formally-real, valued fields to prove a Newton
polygon rule which combines Descartes's rule of signs with the classical Newton
polygon rule.
