Its a well known fact that

Var(X) = E(X^2) - (E(X))^2

but then can you conclude:

Var(X) = E(X^2) - (E(X))^2 = E(X^2) - E( E(X) X ) (because E(X) is a scalar)

= E[X^2 - E(X) X ] = E[ X ( X - E(X) ) ]

by linearity of expectation.

but also Var(X) = E[ (X- E(X))^2 ]

a contradiction no?

Clearly I must be doing something wrong?