The investigator is, in some limited ways, the Pathfinder version of the factotum. Jason Bulmahn, lead designer of Pathfinder, wrote the factotum for Dungeonscape back in the day.
But their inspiration points work in completely different ways, and do different things. The investigator was written for a pure-PF campaign and assumed the factotum with its version of inspiration would not co-exist with it (yes, I know, PF is supposedly backwards-compatible; that claim has been largely forgotten as PF has grown, and definitely is not treated as the default for new material). Despite having the same names, the two should be treated as wholly independent.
“RAW” is already a questionable concept when adaptation is involved (adding PF material to 3.5); by definition for these two things to co-exist, the DM has to be applying houserules to smooth over the gaps. It would be more accurate to say that RAW here doesn’t exist, and that it is incumbent on the DM to fill that gap (and then it would make sense for him to recognize the publication history here and rule they are wholly independent).
However, if you really insist on pretending these problems don’t exist, and want to interpret these pools literally as written, there is no reason to expect them to stack together. Investigator says that it grants a daily pool of inspiration, factotum says it grants a per-encounter pool of inspiration, and the two make absolutely no mention of either each other or the hypothetical existence of other classes with inspiration pools. Unlike psionic power point pools, which explicitly state that multiclassed characters add their pools together, inspiration has no such rule and as such remain separate.
On the other hand, both investigator and factotum state that they can expend “inspiration” to do things. Neither specifies that it comes from the class that granted it. Thus, arguably, selectively-strict-RAW, you could use investigator inspiration to power factotum class features and factotum inspiration to power investigator features.
I have never ruled this way or played in a game ruled this way, and would not consider myself an expert on the investigator class, so I can only speculate on the balance ramifications of this. My gut feeling here is that the best usage of it is factotum 19/investigator 1, effectively granting yourself +Int inspiration points that refresh each day. Considering how very much the factotum loves Intelligence as it is, that could grant the factotum a fair number of points, but the daily refresh rate makes it kind of meh. But it still seems clearly worth more than one more use each for arcane dilettante and opportunistic piety.