matthew---to an important position,but he gave it up for som

could have ...的意思是“本来可以”
should have ...的意思是“应该”
Matthew could have been promoted to an important position,but he gave it up for some reason.
Matthew本来可以升职的,但是他因为某些原因放弃了.
但是用 should have been promoted 的话,也是可以的,只不过后面的句子必须变,比如
Matthew should have been promoted to an important position,but his boss was an ass hole and gave the position to a young girl.
本来是Matthew应该被升职的,但他的混蛋老板把这个职位给了一个年轻女孩.
所以,根据原文句子本来要表达的意思,即Matthew主动放弃,就要用could have been promoted.