Hey rokman54.

From the definition we know Cov(X,Y) = E[XY] - E[X]E[Y] and if Y = X^2 then we get

Cov(X,Y)

= E[X*X^2] - E[X]E[X^2]

= E[X^3] - E[X]E[X^2].

Now if you want you can relate the variance to the above quantity by noting Var[X] = E[X^2] - E[X]^2 where E[X^2] = Var[X] + E[X]^2 where

= E[X^3] - E[X](Var[X] + E[X]^2]

= E[X^3] - E[X]Var[X] - E[X]^3

= E[X^3] - E[X]^3 - E[X]Var[X]

Now Consider E[X^3] and E[X]^3 and the relation to other moments if you want other simplifications.

Did you intend to get the solution in terms of a particular moment of X (like variance, kurtosis, etc)?