Final answer:
In the first quarter of the 17th century, the English East India Company had not set up factories in Broach, Chika Cole (Chikalakere), or Trichinopoly (Tiruchirappalli), as their initial operations were focused on other locations like Surat.
Step-by-step explanation:
The English East India Company was established at the turn of the 17th century and its first factory was set up in Surat in 1608. Even though the company would go on to expand its operations throughout India, such as in Bombay (Mumbai), Madras (Chennai), and other key locations, there is no evidence from the provided references that specific factories were set up in Broach, Chikalakere (Chica Cole), or Trichinopoly (Tiruchirappalli) in the first quarter of the 17th century. The regions mentioned became significant later in history as the company's influence spread across the Indian subcontinent.
By the 1660s and onwards, the English East India Company had a more pronounced presence in several areas, but the factories listed in the question do not align with the trading posts or factories established by the English East India Company during the very early 1600s.