After scaring Pentangeli with his shouting account of the Tahoe shooting, Michael tells Frankie that he .knows Roth was behind it, and asks him to “help me take my revenge.” “Michael, anything…” says the much-relieved Pentangeli. Michael then tells him to “settle these problems with the Rosato brothers.” When Frankie objects, Michael gives him the “keep friends close/enemies closer” rap, then explains that if Roth thinks his relationship with Michael is still good, he’ll be “relaxed…then I’ll find out who the traitor is in my family.”
All good, except that the last thing Michael said to Roth in Miami before leaving to visit Pentangeli was: “Frank Pentangeli is a dead man…you don’t object?” But, instead of having Frankie whacked, Michael dispatched him on a peace mission with the Rosatos. That would have told Roth that Michael didn’t suspect Pentangeli in the Tahoe shooting—meaning that he might suspect Roth. It would make Roth feel anything but “relaxed.”
Why did Michael send Pentangeli on that mission?

I haven't made that connection. Saying Frankie's a dead man mustn't mean it's happening on some set timetable. Until it happens, Michael has to appear to be with Frankie.

how he think I’ll find out who the traitor is in my family.” by sending Pentangeli on that mission?

I find, Michael's visits and the sequence of events somewhat bizarre....

Michael didn't seem to suspect / see through Roth as to why Roth in Miami with no business interests of his own in New York was backing up the Rosato brothers in New York against Pentangeli, Michael's [and Vito's] own loyal, good old man for years until the Tahoe bedroom shooting. Ahhh.... Roth fooled a Corleone but Pentangeli was smarter

• Michael didn't question -- And what is the interest for Roth, in concocting this Love Triangle?!
• did Roth gamble? on Michael's [the heir-apparent] greed for Roth's business that Michael would “give his loyalty to a Jew before his own blood” Roth was not wrong!

Whilst my take, Michael's “Then Pentangeli is a dead man, you don't object.” was a statement rather than a question and Michael was telling [not asking] Roth what Michael was going to be doing ie: Michael was not seeking Roth's approval nor permission

• why did Michael need to visit Roth at all -- because of....
  The important thing? is that nothing interferes with our [Roth and Michael's] plans for the future.
• what is the link between Roth and Michael's plans and the Tahoe shooting
• Roth's been flying under the Radar
  Quote

  Hyman Roth always makes money for his partners.
  One by one, our old friends are gone. Death -- natural or not - prison -- deported.
  Hyman Roth is the only one left -- because he always made money for his partners.
• starting another war by whom
• Roth has never shed any blood never started war/s
• why would the Mighty, Top Don Michael Corleone, visit his subordinate who, according to the Don himself, had just “tried to have Michael killed” In my home! In my bedroom!
• If Pentangeli is small potatoes, how could Pentangeli have engineered such a hit of the Top Don In my home! In my bedroom!
• Did Pentangeli come prepared anticipating Michael's no “to get rid of the Rosato brothers”

After the failed Tahoe shooting and Michael's visit to Roth
1. Neither Michael nor Roth fooled the other
2. Neither was relaxed and confident, in their friendship
3. Both conspiring to get the other killed -- First
4. Both knew the other knew but continued their charade until the First dead man
5. Both underestimated each other, more so

Michael dispatching dead man, Pentangeli on a peace mission with the Rosatos, would make Roth think? his relationship with Michael is still good, he’ll be “relaxed.”

Sure thing Tuna sandwich Picnic!

Why would according to Pentangeli and I believe him [Pentangeli]

Once a common Mafia hood -- Michael needs Pentangeli as his muscle
Otherwise Michael becomes just another casino operator, easy pickings for anyone

• What was Michael expecting? to happen at the Pentangeli-Rosatos meeting
• my take, Michael was not expecting Pentangeli to come to any harm other than the humiliating back down, perhaps handing over the three territories and in the process looking weak etc.
• If Pentangeli is killed by the Rosato brothers - Michael can't afford to let his man's murder go unavenged - danger of starting another war
• Besides if Pentangeli is killed and Rosato brothers, Roth's ally take over the Corleone operation then Michael loses his muscle Pretty soon there won't be one place in Nevada Michael can hang his hat!
• Hence Michael would never want to lose his muscle which is the only thing that makes Michael untouchable

So....

• Why would Michael envisage the Rosato brothers would fail Roth
• and How

It was indeed unbecoming of a Don
1. Michael “undercut Pentangeli through his support of the Rosato Brothers” against his own man
2. Michael gave his loyalty to a Jew [Roth] before his own blood
3. Michael asked Pentangeli to lay down - settle these troubles - to the Rosato Brothers who spit right in Pentangeli's face
4. Michael threw Pentangeli [good old man -- who was loyal to Vito, Michael's father and to Michael, for years] under the bus
5. then abandonment of his 'dead' capo's family

The obvious answer to my question is that it's another of the many director/script "liberties" (or "irrationalities" if you prefer) that FFC and Puzo took to move the film along--in this case to set up Pentangeli's betrayal of Michael, the Senate hearing and all the great drama that followed.

But (and I admit, this is a stretch): Michael, the master manipulator, may have been testing Frankie's loyalty. Sure, he knew Frankie wasn't responsible for the Tahoe shooting. But Michael wanted more: he wanted Pentangeli to put his head into the lion's mouth by meeting with his arch-enemies, the Rosatos. If the mission succeeded, it might have made Roth relax and not get suspicious of why Michael didn't immediately have Pentangeli whacked. Then, as he told Frankie, he'd be able to string Roth along and find out who was the traitor in his family. And then he'd have Roth killed. A very convenient scenario for Michael, but it made Michael relax, and ignore the possibility--perhaps the probability--that the Rosatos would try to kill Pentangeli, leaving his NYC muscle operation in his enemies' hands.

He was slippin'


Turnbull's point raises -- Is this how Michael envisaged the Rosato brothers would fail Roth?
Then the possibility--perhaps the probability--that the Rosatos would try to kill Pentangeli, didn't escape his eye

If the mission failed, leaving his NYC muscle operation in his enemies' hands, then Michael loses his muscle and becomes just another casino operator, easy pickings for anyone. No more Mr. untouchable

If it was the three territories which Pentangeli presumably was handing over, “settle these problems then why kill him?

Why tell Roth Pentangeli is a dead man, then dispatch him?
They would have got them anyway and perhaps more with Roth's backing after Pentangeli is whacked No blood shed as internal disciplinary whacking

Michael put his head into the lion's mouth Both stringing each other along Both saved by Batista's resignation

obvious answer.... that's a cop out!

Sure thing Turnbull there is only so much the director/script can 'story tell' squeeze in, to create all the great drama and to move the film along--

I still find, Michael's visits and the sequence of events somewhat bizarre....
Well, more bizarre than some of the other "liberties" (or "irrationalities") and “plenty of loose ends and inconsistencies”

Great casting in small roles

Why would Michael think the answer to “finding out who was the traitor in his family” was in Havana -- It was great drama indeed among others -

• Michael turning up in Havana without the $2 million
  Roth's gonna think his relationship with Michael is still good That's the key for him!
  Gotta keep Roth completely relaxed and confident, in their friendship
  No little gift for the President…. That would work!
• totally in the dark Fredo bringing the $2 million over
• Fredo denying knowing old man Roth and Ola
• Brothers having drinks together
• Fredo nearly confessing
• Roth's sixth sense
• Giddy! Fredo slipping up and revealing he is the traitor in the family
  We already knew Fredo is the traitor
• distraught Michael's devastation that the traitor is his own brother and crumbling to pieces
• You broke my heart
• Fredo and Michael, one with trepidation and the other broken-hearted
• Party in full swing but the brothers oblivious to the dancing -- uh -- everything
• No words spoken but what each brother is experiencing -- crystal clear

If Geary hadn't asked Fredo "where did you find this place?"