Final answer:
Allen Klein was the businessman who managed both the Rolling Stones and the Beatles. He took over as manager for the Rolling Stones in the 1960s and later for the Beatles towards the end of their career.
Step-by-step explanation:
The American businessman who was first a manager to the Rolling Stones and then a manager for the Beatles is a) Allen Klein. Allen Klein became the manager for the Rolling Stones in the mid-1960s and later managed the Beatles for a short period towards the end of their career.
The other options listed, b) Brian Epstein, c) Andrew Loog Oldham, and d) Peter Grant, are associated with the music industry but do not fit the description of having managed both the Rolling Stones and the Beatles.