Variance of Beta Distribution/Proof 2

Proof
From the definition of Variance as Expectation of Square minus Square of Expectation:
 * $\var X = \expect {X^2} - \paren {\expect X}^2$

From Expectation of Beta Distribution:
 * $\expect X = \dfrac \alpha {\alpha + \beta}$

From Raw Moment of Beta Distribution:
 * $\ds \expect {X^n} = \prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r}$

So:

Therefore: