hi,

so I know E$E\[X|Y$\]=E$X$

how can I do:

E$E\[XY|X$\] ?

I did

E$XE\[Y|X$\]= XE$E\[Y|X$\] = XE$X$
is that right?

&#x200B;

Also in the same vien, how can I handle
E$X+X\^2$ using the law of total expectation?
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) }
I haven't seen E[E[X|Y]]=E[X] and it took me a second to realize the inner expression means E[X|Y] integrated over all Y. It might help if you broke down the other expressions like that to make them easier to understand