Is this way correct:
The density is given by $f(x \mid \theta) = \frac{2}{\theta} x \exp \{-\frac{x^2}{\theta}\}$.
Then, $L(x_1, x_2, \ldots, x_n \mid \theta) = f(x_1 \mid \theta) \cdots f(x_n \mid \theta) = ... = \big( \frac{2}{\theta} \big)^n \exp \{- \frac{u}{\theta}\}(x_1 \cdots x_n) = g(u,\theta)h(x_1,\ldots, x_n)$
with $u = \sum_{i=1}^n x_i^2$. Furthermore:
$\ln(L) = n \ln(2)-n\ln(\theta) - \frac{u}{\theta}+ \ln(x_1 \cdots x_n)$
$\frac{d \ln(L)}{d \theta} = -\frac{n}{\theta} + \frac{u}{\theta^2} = 0$
$\Rightarrow \hat{\theta} = \frac{u}{n} ? \frac{1}{n} \sum_{i=1}^n x_i^2$
Finally, since $E[\hat{\theta}]=\theta$, it follows by factorization theorem and Lehmann-Scheffe, that this is an UMVE.