The Romulan Neutral Zone… providing peak comedy since 2160. 

Interesting story/factoid about this scene: Shatner fought against “Let them die” being Kirk’s (final) reaction.  He fought hard.  To the point that director Nicholas Meyer agreed to film it so that “let them die” wasn’t the final reaction.  If you pay close attention when watching the film, Kirk says the line - they switch to Spock’s reaction - and then they go back to Kirk.  And Kirk, you can tell when you know what to look for, is coming off a different emotion than anger.  He’s coming off of regret:

It’s tough to capture, as its so fleeting; this screen capture doesn’t quite do it.  But it does show a glimpse of the reaction Shatner wanted that Meyer didn’t include in the final cut.  That Kirk says “let them die” full of anger; and then realizes quickly how unworthy and reactionary a response that is and waves his hand dismissive of his own statement. 

In the final film that realization moment is cut in favor of what we see, but I always find it interesting to know these things.  Shatner didn’t like “Let them die” without a counter-balance showing Kirk regretted saying it.

The same was true of a lighter moment in “The Search for Spock” when Sulu flips the security guard and tells him “Don’t call me tiny” - George Takei didn’t like that line at all, and director Leonard Nimoy had to agree to cut it later if it didn’t work in Takei’s opinion to get George to film it.  George did like it in the end, which is why it made it to the film.  But it’s another interesting tid-bit.