In most of the solutions for Prove using mathematical induction that $x^{2n} - y^{2n}$ is divisible by $x+y$ there is a line in which the value $x^{2m}y^2$ is added and subtracted from the equation.
Thereafter the equation is split into two binomials and so on...What is the intuition behind the value $x^{2m}y^2$?
Is there any hueristic for adding 0 in a proof (i.e. adding and subtracting a value)?