How to expose the node creation form in the front page?
To display content type form you can use formblock module as it description explains:


  Enables the presentation of user registration, site wide contact, or node creation forms in blocks. This is particularly useful for including forms on panels.



Enable the module
Go to Project admin/structure/types/manage/project
On the bottom left vertical tabs there'd be a Form Block, check Enable data entry from a block and save
Go to admin/structure/block and enable Project form block




I hope this'd help you.