Fill in the blank in the following sentence with a correct word from…

The correct answer is Option (1) → Further

Further and farther:

• Farther is used for physical distance ("farther from the station").
• Further is used for distance too, and also means more or additional; only "further" has this second sense.
• Deciding clue: "investigation" is not a distance; the case needs more investigation, so "further" is the word.
• Complete sentence: Further investigation is needed in this case.
  Example: For further details, contact the office.

Why the Other Options are Incorrect?

Option 2: Farther — Incorrect. Farther is used only for physical distance; an investigation is not a distance.
Example: The village is farther than it looks.

Option 3: Farthest — Incorrect. Farthest is the superlative of far, meaning at the greatest distance.
Example: Of the three towns, Shimla is the farthest from here.

Option 4: Furthermore — Incorrect. Furthermore is a linking adverb meaning "in addition", used to add a new point to what has been said; it cannot describe a noun such as "investigation".
Example: The plan is costly; furthermore, it is unsafe.