Let X and Y be two random variables, then

Var[X + Y] = Var[X] + Var[Y] - 2Cov[X,Y], where Cov[X,Y] = E[XY]-E[X]E[Y].

I solved this equality and I am still having the following result:

Var[X+Y] = Var[X]+Var[Y]+2Cov[X,Y].

I sent my work to my professor, but he says my work is not correct. Could somebody tell me why? I mean, for which case it is not true?

Var[X+Y] = E[((X+Y) - (E[X]+E[Y]))^2], since Var[Z] = E[(Z-E[Z])^2] is the definition of the variance for a random variable Z.