If $$(1 + xy + x + y)^2 - (1 - xy + x - y)^2 = ky(1 + x)^2$$, then $$k$$ equals to

$$(1 + xy + x + y)^2 - (1 - xy + x - y)^2 = (1+xy+x+y+1-xy+x-y)(1+xy+x+y -1+xy-x+y)$$

Simplifying the RHS by cancelling out terms we get

$$(2+2x)(2xy+2y) = 4y(1+x)(1+x) = 4y(1+x)^2$$

Thus, $$ k = 4$$

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