How do I expand this matrix multiplication with expectations?

Question

I am trying to understand how to expand the following matrix/vector equation with expectation involved: Cov[X] = $E[(X - E[X])(X - E[X])^T] = E[XX^T] - E[X]E[X]^T$

When I expand $(X - E[X])(X - E[X])^T$, do I do it like this: $(X - E[X])(X - E[X])^T = XX^T - XE[X]^T -E[X]X^T + E[X]E[X]^T$ ?

After this, using the properties of expectation, how do I conclude to the RHS?

Answer 1

Notice that the expectation of the term $-\mathbb{E}[X] X^T$ is $-\mathbb{E}[X] \mathbb{E}[X]^T$.