Detection of signal having gaussian noise with non zero mean and a covariance matrix using Neyman-Pearson Lemma(LRT)
You're looking at a Generalized Gaussian Detector$^{\ \text [1]}$. The Neyman-Pearson detector decides $\mathcal H_1$ if:
$$
\frac{p(\mathbf x; \mathcal H_1)}{p(\mathbf x; \mathcal H_0)}&gt; \gamma\tag{1}
$$ 
With $\mathbf w$ and $\mathbf x$ independent; you get here:
$$\mathbf x\sim
\begin{cases}
\mathcal N\left(\mu_w, \mathbf C_w\right) &amp;\text{under} \mathcal \quad \mathcal H_0\\[2ex]
\mathcal N\left(\mu_s + \mu_w, \mathbf C_s + \mathbf C_w\right) &amp;\text{under} \mathcal \quad\mathcal H_1
\end{cases}$$

The NP inequality in equation $(1)$ gives:

$$
\frac{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_s + C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_s - \mu_w\right)^T\left(\mathbf C_s + \mathbf C_w\right)^{-1}\left(\mathbf x-\mu_s - \mu_w\right)\right]}{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_w\right)^T\mathbf C_w^{-1}\left(\mathbf x-\mu_w\right)\right]}&gt;\gamma
$$ 
Taking the logarithm on both sides gives the needed test statistic $T(\mathbf x)$:
$$
\underbrace{\ln\left(\frac{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_s + C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_s - \mu_w\right)^T\left(\mathbf C_s + \mathbf C_w\right)^{-1}\left(\mathbf x-\mu_s - \mu_w\right)\right]}{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_w\right)^T\mathbf C_w^{-1}\left(\mathbf x-\mu_w\right)\right]}\right)}_{T(\mathbf x)}&gt;\ln(\gamma)
$$
You can continue from there. Also, have a look at this related question on NP detector.

$^{\ \text [1]}$: Kay, S. Fundamentals of Statistical Signal Processing: Detection Theory.