**Core Concept**
The patient's presentation of fever, productive cough with green sputum, and a history of bronchiectasis suggests a respiratory infection. The Gram stain of expectorated sputum was performed to identify the causative organism. The clinical presentation and laboratory findings point towards a bacterial infection.

**Why the Correct Answer is Right**
The green-colored sputum and productive cough with a history of bronchiectasis suggest an infection with a bacterium that produces pigments, such as Pseudomonas aeruginosa. Pseudomonas aeruginosa is a common cause of respiratory infections in patients with bronchiectasis and cystic fibrosis. The Gram stain would reveal a Gram-negative rod, which is consistent with Pseudomonas aeruginosa. The presence of pyocyanin, a green pigment produced by Pseudomonas aeruginosa, would also be consistent with this diagnosis.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect as Staphylococcus aureus is a Gram-positive coccus, which would not match the Gram stain findings.
**Option B:** This option is incorrect as Haemophilus influenzae is a Gram-negative coccus, but it is typically associated with acute respiratory infections, such as pneumonia, rather than chronic respiratory infections like bronchiectasis.
**Option C:** This option is incorrect as Klebsiella pneumoniae is a Gram-negative rod, but it is typically associated with pneumonia in immunocompromised patients, rather than chronic respiratory infections like bronchiectasis.

**Clinical Pearl / High-Yield Fact**
Pseudomonas aeruginosa is a common cause of respiratory infections in patients with cystic fibrosis and bronchiectasis, and it is often resistant to multiple antibiotics. Early recognition and treatment of Pseudomonas aeruginosa infections are essential to prevent complications and improve outcomes.

**Correct Answer:** D. Pseudomonas aeruginosa.