Bernstein-Hausdorff representation on an interval

Can every polynomial $P(x)$ of degree $n$ which is real and positive for $-1 < x < 1$ be represented in the form

where $A \geqq 0$, $\alpha + \beta \leqq n$, and $\alpha$ and $\beta$ are non-negative integers?

Every polynomial $P(x)$ which is real and positive in the interval $-1 < x < 1$, can be represented in the form

where $A \geqq 0$, and $\alpha$ and $\beta$ are non-negative integers.