This question tests your understanding of verb patterns after perception verbs like "see". The verb "see" can be followed by either a bare infinitive (without "to") or a present participle (-ing form), but with different meanings.

Step 1: Understand the structure
The sentence is: "Robert saw the doctor ___ the patient."
Here, "saw" is the main perception verb. We need to choose the correct form for the verb "examine" that comes after the object "the doctor".

Step 2: Analyze the options

• to have examined: This is an infinitive perfect form. Perception verbs are not typically followed by this construction. It is grammatically incorrect in this context.
• examine: This is the bare infinitive form. The structure is "see + object + bare infinitive". This form is used to indicate that the speaker witnessed the entire action from start to finish. For example, "I saw him leave" means I saw the complete action of him leaving.
• having examined: This is a perfect participle. It is used to show an action that was completed before the main verb. However, this is not the standard construction after perception verbs like "see". It would be more common in an absolute phrase, e.g., "The doctor having examined the patient, Robert felt relieved."
• being examined: This is a present participle in the passive voice ("to be" + past participle). The structure would be "see + object + being + past participle". This means Robert saw the patient, and the patient was in the process of being examined by the doctor. The focus is on the patient receiving the action.

Step 3: Apply the meaning to the sentence
Let's look at what the subject "Robert" saw.

• He saw "the doctor". Therefore, the doctor is the object of the verb "saw" and is the one performing the action of examining.
• The person receiving the action is "the patient".

We need the construction where the object of "saw" (the doctor) is doing the action. This requires the active voice, not the passive. "being examined" is passive and would make "the patient" the one being seen, which changes the meaning. The most appropriate choice is the bare infinitive examine, which correctly shows Robert witnessed the doctor perform the action of examining.

Final Answer: The most appropriate word to fill in the blank is examine.
The complete sentence is: "Robert saw the doctor examine the patient."

Related Topic: Verb Patterns after Perception Verbs

Verbs of perception (see, watch, hear, feel) can be followed by two structures:

1. Object + Bare Infinitive (e.g., examine): Emphasizes that the entire action was witnessed.
2. Object + Present Participle (e.g., examining): Emphasizes that an action in progress was witnessed; it does not imply seeing the start or end of the action.

The difference is subtle but important. For example:

• "I saw him cross the road." (I saw the complete event from one side to the other).
• "I saw him crossing the road." (I caught a glimpse of him while he was in the middle of the action).

In our original sentence, "Robert saw the doctor examine the patient" suggests he saw the complete act of examination.

Related Formulae

While there isn't a mathematical formula, we can represent the grammatical structure:

$\text{Perception Verb}+\text{Object}+(\text{Bare Infinitive}\parallel \text{Present Participle}(-\text{ing}\left)\right)$