**Core Concept**
The IX (glossopharyngeal), X (vagus), and XI (spinal accessory) cranial nerves are part of the branchial motor and sensory complex, controlling various functions including swallowing, taste, and motor innervation of the larynx and pharynx.

**Why the Correct Answer is Right**
The nucleus ambiguus is a cranial nerve nucleus located in the medulla oblongata that serves as the common origin for the motor fibers of the IX, X, and XI cranial nerves. These motor fibers innervate the stylopharyngeus muscle (IX), the intrinsic muscles of the larynx (X), and the sternocleidomastoid and trapezius muscles (XI). The nucleus ambiguus is a critical structure for the coordination of swallowing and vocal cord movements.

**Why Each Wrong Option is Incorrect**
**Option A:** The nucleus solitarius is a sensory nucleus involved in the processing of taste and visceral sensations from the IX and X cranial nerves, but it is not the common motor nucleus for these nerves.

**Option C:** The dentate nucleus is a deep cerebellar nucleus involved in motor coordination and learning, but it has no relation to the IX, X, and XI cranial nerves.

**Option D:** The red nucleus is a midbrain structure involved in motor coordination and learning, particularly in the context of the rubrospinal tract, but it is not associated with the IX, X, and XI cranial nerves.

**Clinical Pearl / High-Yield Fact**
The nucleus ambiguus is a critical structure for the coordination of swallowing and vocal cord movements, and damage to this nucleus can result in dysphagia and vocal cord paralysis.

**✓ Correct Answer: B. Nucleus ambiguus**