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When does one use 欲しがる instead of 欲しい? For example, in phrases like:

> 先生は野菜を食べて欲し​｛いです・がっています｝。

My incomplete understanding is that the がる form is more formal/polite, but it
can only be used when the subject is not currently present in the room. The
"subject in the room part" is the part I'm most unsure about, as it doesn't
seem right.

Could I say this to my teacher?

> 先生は野菜を食べて欲しがっていますか?

To clarify the above sentence: what I'm trying to say is "Sensei, do you want
me to eat vegetables?"
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grammar politeness i-adjectives
When to use 欲しがる instead of 欲しい
Quoting [an answer from rintaun to one of my own
questions](https://japanese.stackexchange.com/questions/1520/is-suffix-
limited-to-specific-adjectives-only/1523#1523):

> ～がる is a suffix for representing a third party's apparent emotion.

So I would say the major difference between 欲しがる and 欲しい is that while 欲しい is
the state of having desire, 欲しがる is the act of expressing it and making it
apparent, like making intense face, licking your drooling lips etc.

EDIT: To address the example sentence `先生は野菜を食べて欲しがっていますか?`, casting the lack
of keigo aside, I would say the question is only okay to ask a third person,
like for example when your teacher is visiting your home and you ask your
wife, "Does Mr. Smith look like he wants to eat vegetables?". You wouldn't ask
that same question of the actual person, "do you look like you want to eat
vegetable?" because that would seem like asking him to look at a mirror (i.e.
sarcasm). You would simply ask "do you want to eat vegetables?",
`先生は野菜を食べて欲しいですか?` However, considering how Japanese respects teachers, that
would still be too blunt, so you would need a more considerately crafted
question like "Would vegetable be fine with you?", `先生は野菜でよろしいでしょうか?`
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Japanese has a curious unwritten rule which states, in essence, that you
cannot presume to know the intimate details of a third person's mental state.
This is quite an unfamiliar concept in English-land:

> ○ 私【わたし】はDSが欲【ほ】しいです。 I want a DS.
>
> × 息子【むすこ】はDSが欲【ほ】しいです。 My son wants a DS. _(OK in English, NG in Japanese)_

Even if your son has been begging you for the last six weeks straight to buy
him one of them newfangled DS things (because all his friends have one, even
Kentarō, and his parents never buy him anything) and even if you're 100%
certain he wants a DS, you can't say this directly. The ～がる suffix is for
getting around this problem.

> 息子【むすこ】はDS **を** 欲【ほ】しがっています。 My son wants a DS. _(note the particle
> change!)_

There are other ways to escape this problem without resorting to ～がる, each
with slightly different nuances.

> 息子【むすこ】はDSが欲【ほ】しい **と言【い】っています** 。 My son **is saying** he wants a DS.
>
> 息子【むすこ】はDSが欲【ほ】しい **ようです** 。 **It would seem** my son wants a DS.

Now for your example:

> Could I say this to my teacher?
>

>> 先生【せんせい】は野菜【やさい】を食【た】べて欲【ほ】しがっていますか?

No. ～がる is for a third person's emotions, so you cannot use it when you are
talking directly to someone as here. But if we remove ～がる, we end up with
this:

> 先生【せんせい】は野菜【やさい】を食【た】べて欲【ほ】しいですか。 Do you want (someone) to eat their
> vegetables?

I'm guessing this is not what you wanted to say. Remember that the ～て欲【ほ】しい
construction refers not to something the subject (in this case, 先生【せんせい】)
wants to do, but something the subject wants _someone else_ to do _for_
him/her. The agent (the entity actually performing the action) can be marked
by に:

> 先生【せんせい】は生徒【せいと】に野菜【やさい】を食【た】べて欲【ほ】しいですか。 Do you want the students to eat
> their vegetables?

But I'm guessing you wanted to ask your teacher if he/she wants to eat some
vegetables. In that case, you could use the ～たい form (again without ～がる).

> 先生【せんせい】は野菜【やさい】を食【た】べたいですか。 Do you want to eat some vegetables?

Although this is grammatically correct, in this case it is _socially_
incorrect. It's inappropriate in Japanese to directly inquire as to the
desires someone of a higher status like this. A better way to phrase this
would be:

> 先生【せんせい】は野菜【やさい】はいかがでしょうか。 Would you like some vegetables?
>
> 先生【せんせい】は野菜【やさい】でよろしいでしょうか。 Would vegetables be OK? _(suggested by Lukman;
> implies alternate choices besides vegetables)_
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Whenever you're talking about someone else's internal feelings, you use
欲しがっている rather than 欲しい, as Derek Schaab beautifully described in his answer.
To add a little more background, there are other words that this is done for
other than 欲しい:

> [verb]-たがる (want to do some verb)  
> 痛がる hurt/painful  
> 嬉しがる happy/glad
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