= Solution
Set $R=1+r$, $a=\mu-RS_0$, and $M=(\gamma V)^{-1}$. The terminal wealth is $W_1=Rw_0+\theta^T(S_1-RS_0)$, a <normal random variable> with mean $Rw_0+\theta^Ta$ and variance $\theta^TV\theta$. Its <moment-generating function> gives
$$
\mathbb E[-e^{-\gamma W_1}]=-\exp\left[-\gamma(Rw_0+\theta^Ta)+\frac{\gamma^2}{2}\theta^TV\theta\right].
$$
Maximizing this expected <exponential utility> is equivalent to maximizing the strictly concave <quadratic form> $\theta^Ta-(\gamma/2)\theta^TV\theta$. Since the nonsingular covariance $V$ is positive definite, the unique stationary point is the unique maximum:
$$
\boxed{\theta^*=M(\mu-RS_0),\qquad x^*=w_0-S_0^TM(\mu-RS_0).}
$$
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