exponential sufficient statistics 
A family of pdfs is called an exponential family if $$f(x|\theta) = h(x)c(\theta) \exp \left(\sum_{i=1}^{k} w_{i}(\theta) t_{i}(x) \right)$$ and the statistic $T$ is sufficient iff $f(x;\theta) = h(x)g(T(x);\theta)$ for two nonnegative functions $h, g$.

How can I use this to show if this is an exponential family and find a sufficient statistic for $\theta$ if $X_1, X_2,..., X_n$ are a random sample from this density?
$$ f_{\theta} (x) = \frac{6x(\theta - x)}{\theta^3} $$$$ 0<x<\theta$$

I don't think this distribution family is an exponential family because one can rewrite the density according to Lebesgue measure (I assume your model is dominated by Lebesgue measure) as :
$$\forall \theta \in \Theta, ~~ f_\theta(x) = \frac{6x(\theta-x)}{\theta^3} \underbrace{\mathbb{1}_{[0;\theta]}(x)}_{\textrm{can't be rewritten as a(x)b(}\theta)}$$
