Since you haven't supplied any initial condition for your DE there will be an unknown constant in the general solution
f = FullSimplify[P[x]/.DSolve[P'[x] == 3 P[x] - 2 P[x]^2, P[x], x][[1]]]
which gives you
3/(2 + E^(-3 x + 3 C[1]))
You can peek at a plot for a handful of values picked for the constant C[1], but C[1] doesn't have to be an integer, and see the behavior
Plot[{(f/.C[1]-> -2), (f/.C[1]-> -1), (f/.C[1]-> 0), (f/.C[1]-> 1), (f/.C[1] -> 2)}, {x, -10, 10}]