What is the significance of the quote, "Well I've had a very bad time...and I'm pretty cynical about everything."
In The Great Gatsby, following the quote you mention, Daisy shows Nick a bit of herself that she doesn't usually reveal. She tells Nick how disappointed she was when she gave birth to a daughter, instead of a son. Daisy shows weakness and despair. She says:
"It'll show you how I've gotten to feel about--things. Well she was less than an hour old and Tom was God knows where. I woke up out of the ether with an utterly abandoned feeling and asked the nurse right away if it was a boy or a girl. She told me it was a girl, and so I turned my head away and wept. 'All right,' I said, 'I'm glad it's a girl. And I hope she'll be a fool--that's the best thing a girl can be in this world, a beautiful little fool.'
Daisy uses this anecdote to explain why she is cynical. And remember that when she says:
"...I've had a very bad time, Nick, and I'm pretty cynical about everything"
Evidently she had reason to be.
Daisy shows despair, here. Her daughter--born into a male-dominated world--will face what women have faced for centuries: an existence in which the best and sometimes the only way to improve oneself socially and economically is to marry a wealthy man. That, of course, is what Daisy has had to do.
Though Daisy seems to play her despairing revelation off with a "smirk," and Nick interprets the anecdote and revelation as an act, it is difficult to not see at least some truth in it. Especially when one considers Nick as an unreliable narrator projecting his Midwestern values upon everything he narrates. At least some of Daisy's despair is real.
Remember, when Daisy ultimately rejects Gatsby, she does so because he asks too much, she says, because he wants her to say that she has pined for him for the last five years and that she never loved Tom. And she refuses to do that. She does not reject Gatsby out of greed--Gatsby has plenty of money, and certainly has more style than Tom.
This quote is from Chapter 1 and it is spoken by Daisy. She is talking to Nick when she says this. To me, the quote is significant because she tells us something very important about her character. She says she is cynical, and the cynicism is going to be a big deal in this book.
Her cynicism is going to be important because she is going to treat Gatsby in a very cynical way. He has all these idealistic feelings about her, but she does not respond in kind. The climax of the book will come when we see her respond cynically to Gatsby's love.