Step 1: In a debilitated, catheterized inpatient with both a urinary and a respiratory infection, the likely pathogens frequently produce beta-lactamase enzymes that destroy simple penicillins.
Step 2: The rational pick is a beta lactam paired with a beta lactamase inhibitor; the inhibitor shields the antibiotic, restoring its broad activity against these organisms.
Step 3: This combination therefore gives potent, dependable coverage for the UTI and pneumonia together.
Step 4: Unprotected amoxicillin would be inactivated, and the second- or third-generation cephalosporin choices are less optimal for this resistance-prone setting.
\[\boxed{\text{Beta lactam with a beta lactamase inhibitor}}\]