Given the parametric class formed by the density functions defined as follows: $$p(y;\theta)=\theta(\theta+1)y(1-y)^{\theta-1},\ y\in(0,1),\ \theta >0$$ Does this parametric class forms an exponential family like $$f(y, \theta)=q(y) exp\{\phi(\theta)t(y)-\tau(\theta)\}$$ ?
I could write the density function in this way:$$p(y;\theta)=y\exp\{(\theta-1)ln(1-y)+ln(\theta(\theta+1))\}$$
so, if
$$q(y)=y,\ \phi(\theta)=(\theta-1),\ t(y)=,\ \tau(\theta)=-ln(\theta(\theta+1))$$
then the answer should be yes. Is that correct?
Is also correct the relationship $p(y;\theta)=Beta(2,\theta)$?