Final answer:
The characteristic not shared by both living organisms and cells is 'c) Grow and die,' as cells typically divide and do not die as part of growth, while organismal death is the end of all biological functions.
Step-by-step explanation:
The characteristic that is not shared by both living organisms and cells is c) Grow and die. While both cells and organisms grow, cells do not die in the same way multi-cellular organisms do. The cell cycle includes growth, but cells typically divide rather than die as part of normal growth for multi-cellular organisms. The process of cell death, or apoptosis, is a regulated function within an organism's life cycle. Comparatively, when an organism dies, it is the cessation of all biological functions that sustain its life, which is distinctly different from the regulated cell death within living organisms.
Here's why the other options do not fit as the exception: a) Remove wastes from themselves, b) Rely on chemicals for communication, and d) Extract and use energy from food are all functions that both cells and entire organisms carry out. Both need to manage waste to maintain homeostasis, both communicate using chemical means (such as neurotransmitters or hormones in multi-cellular organisms and signaling molecules in single cells), and both need energy from food, which cells gain through processes like cellular respiration using nutrients like glucose and lipids, potentially produced by photosynthesis or ingested.