Yes. He does if you watch the DBZ Buu Saga when he going to die.
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Vegeta
Yes but trunks is half saiyan and half human since his dad vegeta is from planet vegeta and Bulma his mom is from earth
If you're talking about during the Buu saga, when Vegeta decided to sacrifice himself, then you're half right. Vegeta hit Trunks across the head and punched Goten in the stomach to knock them out so they wouldn't interfere.
he has 2 kids and their names are Trunks and Bulla
Yes. Vegeta does care for both Bulma and Trunks. He willingly sacrifices himself to Buu for them. But it doesn't work. I guess it's the thought that counts. and we even see Bulma crying after she hears the news.