Written approval? Wow! Okay then, I would start by considering the benefit to their gardens, though most people know that better pollination will be beneficial, you might provide some statistics for your gardening neighbors of what percentage garden yield they might expect if you have bees.
I would then move on to calming what their concerns might be. Concerns include flight paths (so explain how you will train bees to go up and over), water collection (how you will provide water supplies in your own yard), whether or not the bees will react to them mowing their lawns and attack them (unless the hive is being directly threatened, not the way of the bee), swarming (explain the docility of swarms and how to contact you if they see one and promise to be there right away or send someone right away).
And anything else you can think of. I think it assures people if you have an immediate answer for all their questions. It's when you come across as unsure that people let their own fears get out of hand.
And don't forget the old fashioned method of bribing. A small share of any honey collected to those you needed written approval from is always nice. You can even provide a sample to them from another beekeeper now to provide an example of future harvests.