I am wondering if my understanding of these two phrases is correct. Could you confirm this?
- make a contract = create a contract
- achieve a contract = win a contract
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For no. 1, more acceptable phrases are:
enter into [a] contract Example: We must enter into a contract with XYZ Corporation if we want to generate more sales during the next quarter.
contract with someone (for something) and contract (with someone) for something to make an agreement with someone to produce or supply something, or to do something. I will have to contract with an expert for that part of the project. We contracted with a local builder for a new kitchen. Did you contract for plumbing work with Eric? http://idioms.thefreedictionary.com/contract
For no. 2, "win [a] contract" is already good or correct. It is more commonly used than 'achieve a contract'. Example: I am confident we will win this contract with ABC Company.