“Would you consider going into business with me?”
You're going to start a gourmet catering business. You want your friend to join you as a business partner. You told her about the kind of business you want to start, and she seemed interested in it. You ask her this.
Would you consider going into business with me?
You ask "Would you consider ___ing?" when you want to carefully and politely propose an idea to someone. You usually use this phrase when asking about big and important decisions. For example:
Would you consider moving in with me?
Would you consider coming over to work for us?
To "go into business with" someone means to start a new business with them or to join them as a partner in the business. A "partner" is usually someone who gets a % of the business, not just a salary.