你这句话用that是不合适的,最好用which引导的非限制定语从句才行.我先把这句话改了：
The passage briefly tells us that a family, which had eaten fake food, were found by a neigbor and sent to a hospital at once.
限制性定语从句和非限制性定语从句的区别：在限制性定语从句中,如果从句去掉了,这句话就不成立了；在非限制性定语从句中,如果从句去掉了,虽然意思不完全,但句子依然很完整.这就话就是的,如果去掉which had eaten fake food,虽然意思可能不充分了,但句子还是很完整的.
你这句话还有两个错误, neighor和hospital之前一定要加上冠词才行,因为这两个词是可数名词.