Let

a = x+y

b = xy

Then

(x+y)(x+y+2xy)+(xy+1)(xy-1)

is transformed

a(a+2b)+(b+1)(b-1)

=a^2+2ab+b^2-1

=(a+b)^2-1

=(a+b+1)(a+b-1)

Therefore

( x+y)(x+y+2xy)+(xy+1)(xy-1)

=(x+y+xy+1)(x+y+xy-1)

=[(x+xy)+(y+1)](x+y+xy-1)

=(x+1)(y+1)(x+y+xy-1)