F.4 AM

Let P(n) = (x^n)-(nxy^n-1)+[(n-1)y^n]
for n=2
P(2)=x^2-2xy+y^2=(x-y)^2
therefore P(2) is divisible by (x-y)^2
then assuming P(k) is divisible by (x-y)^2, hence we have
x^k -kxy^k-1+(k-1)y^k =M*(x-y)^2 -----(1)
for n=k+1
P(k+1)=x^k+1- (k+1)xy^k + ky^k+1
=x[M*(x-y)^2 + kxy^(k-1) - (k-1)y^k] - (k+1)xy^k +ky^(k+1)
=M*(x)(x-y)^2 + kx^2y^(k-1)- (k-1)xy^k -(k+1)xy^k+ky^(k+1)
=M*x(x-y)^2 + ky^(k-1)[x^2+y^2] - 2kxy^k
=M*(x)(x-y)^2 + ky^(k-1)[x^2+y^2 -2xy]
=M*(x)(x-y)^2 + ky^(k-1)(x-y)^2
=(x-y)^2[M*x+ky^(k-1)]
therefore P(k+1) is divisible by (x-y)^2
hence, by M.I
P(n) is divisible by (x-y)^2 for all positive integers n greater than 1