The sentence gives three words, expedite, hasten and hurry, and asks for a fourth word that carries the same sense. All three words are about speeding up an action, so the blank needs a word with that same speed related sense. Let's go through each choice.
Only "accelerate" keeps the same idea of speeding a process up, so it is the word that fits the blank.
Let's summarize:
So the correct word for the blank is accelerate.