TUYMAADA 2020 S8

The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$ . These polynomials satisfy the identity $$P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1.$$ Determine all possible values of $Q \left( - \frac{1}{2} \right)$ .