probability - Proove that Var Y=$nalphabeta /(alpha+beta)^2$

Let $(X_i, P_i)$, with $_i$ being an integer be an independent random variable s.t. $X_i|P_i$ ~ $Bernoulli(P_i)$ and $P_i$ ~ $Beta(\alpha, \beta$.

Here's what I have so far:
Var Y=Var(E(Y|P)+E(Var(Y|P) is what we know as our formula;

If we break down the two components on the right side we get:
Var(E(Y|P))=$\alpha\beta/(\alpha+\beta)^2 (\alpha+\beta+1)$ and
E(Var(Y|P)=$E(P_i(1-P_i))=\alpha/(\alpha+\beta) (1-\alpha/(\alpha+\beta))$.

Somehow I'm not getting the right result.

Is this correct and if not, could you please correct my understanding?