Krandick's proof of Lagrange's real root bound claim

In 1879 Lagrange asserted without a proof that a certain sum of two radicals would always be an upper bound on the positive roots of any polynomial having a positive leading coefficient and at least two negative coefficients. The claim was nearly forgotten and never proved. This paper provides a nontrivial proof by Krandick and his application of the resulting theorem to an improvement of the very effective Hong root bound.