It's common for n to denote a fixed object, e.g. the nth version of something. m is a variable.
If so, then say n = 2 in both P1 and P2.
The number of terms in P1 increases by 1 each time: (ax+by), (ax+by)^2, (ax+by)^3 etc., as m increases.
However, the number of terms in P2 starts smaller, but increases more quickly: (ax)^2, (ax+by)^2, (ax+by+cz)^2, (aw+bx+cy+dz)^2 etc., with the length of the expression corresponding to the triangular numbers.
So the number of occasions on which P1 is longer is finite, but the number of occasions on which P2 is longer is infinite.
Posted by broll
on 2014-07-16 12:29:36