could you please explain the following:

She was very concerned about the speeches he was making and dangled carrots in front of his nose to try to shut him up.

so what does dangled carrots stands for, also to shut him up?

to dangle carrots in front of your nose...

this is typically associated with donkeys/mules. People used to dangle or hang a carrot in front of the nose of a donkey to get it motivated. The donkey would walk towards the carrot in order to eat it but as the donkey moved, the carrot also moved, so the donkey would continue to try and get the carrot, which in effect would be making the donkey walk. (Donkeys can be stubborn sometimes and not want to go where you want then too)

SO..the idiom means: to offer something to someone in order to tempt them to do something.

To shut up = to make someone STOP talking.

In your sentence, the Woman is tried to make the man stop speaking by offering him something (maybe a distraction).
Well, let us take credit card companies, for example. In order to market themselves, they could dangle carrots by promoting initial low interest rates, which would attract customers.
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I would say she offered him carrots as a treat in order to make him stop talking. What a weird sentence! I suppose it means sth like trying to influence you in some way. Why would anybody do such a thing? Maybe Hitch could provide us all with a rational explanation.
 Woodward's reply was promoted to an answer.
do you have any more examples i find this one particularly difficult to understand?
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 Jacko's reply was promoted to an answer.
I like this example a lot better. Thanks.
I see that couple in a formal social meeting, maybe the first day the parents meet the boyfriend of hers. The boy is saying nonsense and she doens't know how to change his speeches, she is bringing new topics (the carrots) to the conversation trying he stops.
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This is a good example Jacko. I just realised that I didn't give an example, only a type of explanation.