'quickly' - well that's subjective
The pool has to hash the coinbase transaction that they make up part of, hash the side of the merkle tree to get the merkle root, hash the block header that includes the merkle root, then you have a block hash which of course has a resultant difficulty.
Thank you, ck explained it to me. What I didn't realize was the pool has to the full calculation as well ... the difference is it only has to verify the share is legit. The miner's job is the find the legit share.
M