You have the "law" right (in scare quotes because this is just part of the definition of conditional expectation in measure-theoretic probability). But you are messing up the rest. You can pull X out of E(X Y | X) but you cannot pull X out of the *unconditional expectation. Similarly, the "law" is not going to help you with E(X + X^2). If you cannot do E(X Y) and E(X + X^2) without any reference to conditional expectation, then you cannot do it with.

It is true though that if you know E(Y | X) that E(X Y) = E{ X E(Y | X) }