How to search with partitions in my query?
As per my understanding, you are using a function in the where clause. That's why the explain plan is not able to identify the exact partition number in table's partitions.

Function You used : (date_sub(now(), interval 2 day))

For using index: you have to create an individual index for that column like 

create index index_tei_trn_date on transaction_event_items(tei_trn_date);


If explain plan still doesn't use the index, use hint index. For more details on this read: http://dev.mysql.com/doc/refman/5.1/en/index-hints.html