enquiry or inquiry: According to Fowler, inquiry should be used in relation to a formal inquest, and enquiry to the act of questioning. Many (though not all) British writers maintain this distinction; the OED, on the other hand, lists inquiry and enquiry as equal alternatives, in that order. Some British dictionaries, such as Chambers 21st Century Dictionary, present the two spellings as interchangeable variants in the general sense, but prefer inquiry for the "formal inquest" sense. In the US, only inquiry is commonly used. In Australia, inquiry and enquiry are often interchangeable, but inquiry prevails in writing. Both are current in Canada, where enquiry is often associated with scholarly or intellectual research.
As there is a clearly defined objective, inquiry would be correct. The police, in this case, are seeking to find out why and by whom these persons were killed, as they already know how.
Enquiry, should be used, IMO, where there is no clear objective, and the questions are merely to find out.