The next Labor of Hercules was to go to King Augeas, who assigned him to go to his stables, kill ten rats, and bring him their tails. Erm, excuse me.  King Augeas was the proud owner of a herd of 3000 divine cattle, which he kept in a stable, which he never cleaned. In this way, he is often credited with the invention of factory farming. Augeas, so convinced that this nasty, demeaning job would conquer Hercules, bet him a tenth the herd that he could not do it in a single day.
Hercules diverted two rivers, cleaned out the stables in an hour or so, then went to claim his reward, whereupon we learn the moral of the story: don't mess with a naked bodybuilder. Augeas didn't pay up, Hercules killed him. A lot of people wind up dead around Hercules -- isn't that how he was assigned the Labors in the first place?
Hercule Poirot takes on the modern day equivalent of the Augean Stables: Politics. A popular prime minister has recently stepped down, and his son-in-law (by all accounts a "sound man") has taken office. Almost immediately, a tabloid gets wind of scandal, and manages to prove that the previous prime minister was a total crook. The new guy looks into it and discovers that it's all true. He and one of his ministers come to Poirot for help. Poirot seriously considers whether to take this case.
What follows is more of a caper story than anything else. Poirot attempts bribery somewhat half-heartedly. (And here Christie breaks the fourth wall in a way she very rarely does, remarking that this refusal was his excuse to do what followed)
Most of the rest of the story is in a very broad view: no characters as such, just very short snippets of reaction to the news coming out. We get a surprise -- the allegations about the Prime Minister take a back seat to allegations of his wife's improprieties. There are pictures of her with some lothario, and there are rumors of all manner of debauchery until finally she has a nervous breakdown!
Her husband, the current (good) PM sues for libel. A star witness is produced who completely refutes the charges. Now nobody believes anything the tabloid says, and it folds. Poirot emerges at the very end, and explains that he was behind the false charges -- he swept away the muck with a different force of nature: sex.
On a personal level, I vacillate on whether I like this story or not. On the one hand, it's well-written and exciting, and a refreshing change of pace from the other stories. On the other hand, it's not very satisfying. True, he allowed the tabloid to do itself in: if it had not been greedy and salacious, it would have never fallen prey to Poirot's trick. But it drives me nuts that he didn't suggest the simple expediency of coming clean, beating the tabloid to the punch.
Anyway, from a writing point of view, I was struck by the sort of zoom effect: it starts focused tightly on Poirot, then completely zooms out for an omniscient view of public reaction as it evolves over weeks. Then it zooms back in.
I have seen this technique before, in works that I thought were not well done. Here, I think it works primarily because it is coupled with a change in style. The writing becomes choppy, abrupt. Several short segments, each starting with, "People were talking." In each we get snippets of nameless people talking, showing how these representative samples are focusing on the salacious details and leaving alone the (true!) political matters. It works well, but only because it is short. The alternative, focusing on Poirot, would have been weaker: he would have betrayed himself to the reader, for one thing, but he would also merely be a filter between the reader and the public whose reaction is at the heart of the story. (There's also a reprise of the "no smoke without fire" theme from earlier, which I thought a nice touch)
I'm also struck by how hard Christie works to make the prime minister an obvious "good guy": oh, he wants honesty, he mocks himself for being just another politician. This is where his companion is useful: the other minister (who wants to hush it up) is an excellent contrast. Tom DeLay would look moral and forthright next to this guy. Oh, and the fellow who would win the election if this government fell would be most unsuitable! Out of the question.
In the end, it's not a bad story, but I think it's fatally weakened by not addressing what I feel is the obvious solution, that of coming clean. By not addressing what looks like an obvious potential solution (it need not be done, just discounted) I felt a little like the whole rigmarole was a bit of a farce. More than that, I felt it broke Poirot's character a bit.
Lessons learned for me: brief periods of "clever" reaction writing can work well. If you're presenting the reader with a Gordian knot, clean the area of would-be Alexanders. (It need not be too explicit, just enough to let a reasonably intelligent reader discount it)
 But seriously, I've been struck several times by how similar the Labors of Hercules are to modern RPG quests. "Go here, kill this thing, come back." All it needs is pixellated graphics, a repetitive soundtrack, and a 3/4 perspective and you've got half the RPGs ever made. Most platforms would require the characters to wear more clothing, though.