An upper bound for the roots of X-d + a(1) Xd-1 + ... + a(d-1) X + a(d) is given by the sum of the largest two of the terms This bound by Lagrange has gained attention from different sides recently, while a succinct proof seems to be missing. We present a short, original proof of Lagrange's bound. Our approach leads to some definite improvements. To benefit computationally from these improvements, we construct a modified Lagrange bound which at the same asymptotic computational complexity is at most 11 per cent from optimal for degrees d >= 16. (C) 2017 Elsevier Ltd. All rights reserved.