Explain why an x-ray of long bones can determine if someone is done growing. what would a doctor be looking for?

It depends on the situation. For example, if a person is 21 or 22 years old, they are expected to have finished growing and such an X-ray is not needed (don't want to irradiate people for no good reason). If a person is young, such as before puberty or in early puberty, and we want to guess how many years are left to grow taller, we can do an X-ray to see if the growth plates are closing or not.

Bone age determination. Bone age is helpful in predicting adult height in early- or late-maturing adolescents and x-ray of hand and wrist is typically utilized in determining the same. For instance, most girls stop growing when they reach age 16. However, girls 16 years or older have a potential to grow further if their bone age are lower than their chronological ages.

The long bones like your legs contribute the most to final height while vertebrae contribute less. Growth plates are located in back of the end of bones & add new bone over time. They appear on x-ray as thin lines not as dense as bone. Puberty sends out hormones that speed & finally ends growth. When the less dense lines become disappear, the growth plates have closed, ending the growth process.