Bat Par'oh found a three-month-old Hebrew baby in/around the Nile (Shemot 1:5-10) and had him fed and raised by Yokheved until approximately age 2 (Shemot Rabbah 1:26). Bat Par'oh then raised this baby presumably with her father's knowledge.
So his daughter comes home with a random two-year-old boy who presumably looks highly un-Egyptian (after Par'oh's astrologers already told him they were uncertain if the Hebrews' redeemer was Egyptian or Hebrew according to RaSh"I on Shemot 1:22). Given the psychotic behavior the Torah attributes to Par'oh, did it not occur to him to ask where his daughter brought this boy from?
In short, did Par'oh know Moshe was a Hebrew while Bat Par'oh was raising Moshe? If so, why didn't Par'oh put two and two together and have him killed? If not, how do the commentators deal with why not?