My daughter got a Hyacinth at the store. I don't know if it's safe to trim the roots if they get too long for the jar.?

NO - don't trim the roots.

In this case, it means "forever"means one season, and through its flowering period. Spring-flowering bulbs, even in their natural habitat, come up in the spring, and fade aways and disappear entirely by summer, They're only "awake" for a very short time each year.

You have a "forced" spring flowering bulb there, and it will only look good for a few weeks. Forced bulbs rarely bloom again the following year - even it you DO give it nutrition, sun, and the 2-3 month chilling period required to do so. Most people treat these as annuals, and toss them out after they're done flowering.

If you like, and you live where there is winter, plant the bulb in the ground, about 4 inches down, when the leaves start to yellow. It will go dormant and disappear from the surface, but reappear next spring. It probably won't bloom next year, but it should send up leaves, and with luck, will bloom the following spring. (And if you live where there is no winter, or winters are very mild, don't bother - it will never re-bloom without a 2-3 month chilling period every year. Just throw it into the compost when it no longer looks good)

Here's a picture of the roots.

No. They don't require trimming. It will begin to fade away after flowering and with no nutrition or proper sunlight the bulb won't have enough energy to flower next year.

Put the plant in the soil in a garden in the spring and do not let it get sunburn. It will adapt.

if they get too long after blooming, just cut the top of stem to promote the growth of root

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