response of LTI system to a Random Input Signal
There are some basics that you really need to understand. The expectation operator of a deterministic (non-random) function just equals the function. Since the impulse response $h(t)$ is considered to be deterministic, you have

$$E[h(\tau)X(t-\tau)]=h(\tau)E[X(t-\tau)]$$

Furthermore, since $X(t)$ is known to be stationary, its mean doesn't depend on time, so you have

$$E[X(t-\tau)]=E[X(t)]=m_x$$