This question tests your understanding of verb patterns and sentence structure in English. The correct answer is determined by analyzing the grammatical construction needed after the verb "have" and the appropriate form following a preposition.

Step 1: Analyze the Main Verb
The sentence describes an action Richard is performing. The main verb is "have". In this context, "have" is used to mean "to cause something to be done" or "to experience something being done". The correct pattern is "have + object + base form of verb". Therefore, we need the -ing form "having".

Step 2: Analyze the Second Blank
The word "the doctor" is the object. After this object, we need the base form (infinitive without 'to') of the verb "examine" to complete the causative structure "have someone do something".

Step 3: Evaluate the Options

• having / examine: This follows the correct pattern "have + object + base verb".
• being had / to examine: "Being had" is a passive construction which doesn't fit the active voice of the sentence. "To examine" uses the infinitive form, which is incorrect after the causative "have".
• to have / to examine: "To have" is an infinitive, but the sentence requires a continuous action ("is..."). "To examine" is also incorrect after the causative "have".
• to be having / examine: "To be having" is an infinitive continuous form, which is not grammatically correct following "is".

Final Answer: The most appropriate words to fill in the blanks are having and examine.
The complete sentence is: Richard is having the doctor examine his chest.

Related Concepts & Formulae

Causative Verbs (have, get, make, let):
Causative verbs indicate that one person causes another person to do something. The structure for "have" is:
Subject + have + object + base form of verb
Example: I had the mechanic fix my car.

Verb Patterns:
Understanding which verb form (base, infinitive, gerund) follows another verb is a key grammar skill. Some verbs are followed by the base form, some by the infinitive (to + verb), and some by the gerund (-ing form).