The correct option is A started
Options A and B: 'Began' and 'started' mean the same thing, but their usages are slightly different.
'Begin' means that 'you're performing the first part of an action' whereas 'started' means 'to cause an event to happen'.
When referring to a business, you always 'start a business' and not 'begin a business'.
Hence option A is incorrect and B is correct.
Option C: 'Did' refers to performing an action, it does not agree here as 'business' is not an action.
Option D: 'Hoped' means 'to wish for or expect something to happen'. As a result the sentence sounds vague here, it requires a preposition.
Hence options C and D are incorrect.