It is possible for a cell to make proteins that last for months; for example, hemoglobin in red blood cells. however, many proteins are short-lived and may be degraded in days or even hours. why do cells make proteins with such a short life?

No, it is not possible for a cell to make proteins that last for months because; the functional proteomes function effectively if they are short-lived the protein homeostasis and turnover needs to be maintained.

Explanation:

In most of the eukaryotes including humans, cell life cycle follows an internal body clock as per its requirements. Depending upon this clock, the cell and its components like proteins survive or degrade.  Protein half-time depends upon cell life cycle. Protein molecules are constantly and regularly synthesized, degraded, and replaced to maintain protein homeostasis and turnover. Proteins are short-lived may be degraded in days or even hours to aid the cells to activate and control all the cellular activities in a more precise, efficient, and rapid manner. However, exceptionally, certain proteins which mature over age like crystallin present in the human eye lens, dentine in tooth, and elastin, and collagen are long-lived .