Csdrhrt

So I'm trying to derive an expected value (related to Bayesian risk/loss function) and I've derived all except one final part. To finish the final part I need to derive one of the following expected values (either will work)

Define the probability mass function

$$p_N (n) = {n-1 choose x-1} frac{Gamma(a+b)}{Gamma(a) + Gamma(b)}frac{Gamma(a+x)+Gamma(b+n-x)}{Gamma(a+b+n)}$$

for $n =x,x+1,x+2,dots$ and also define the conditional pmf

$$p_N(n|p) = {n-1 choose x-1} p^x (1-p)^{n-x}$$

To complete the final step I need either one of:

$$E_N left[ left(frac{x-1}{N-1} right)^2right]$$

or

$$E_{N} = left[ left(frac{x-1}{N-1} right)^2 bigg| p right]$$

In previous questions, I've derived the necessary expected values by absorbing the terms into the probability functions in order to construct a new distribution function, and then obtaining the expected value by normalizing it so that it sums/integrates to $1$. But for these ones I'm stuck due to the fact it's squared and you're left with a single fraciton that can't be absorbed into the combination.

Does anyone see a way forward?