Question 21
The polynomials $P(x)$, $Q(x)$ with real coefficients are non-constant, monic and satisfy the equality:
$$
2P(x) = Q\left(\frac{(x+1)^2}{2}\right) - Q\left(\frac{(x-1)^2}{2}\right), \quad x \in \mathbb{R}, \quad P(1)=1,
$$
Determine the polynomials $P(x)$ and $Q(x)$.
$$
2P(x) = Q\left(\frac{(x+1)^2}{2}\right) - Q\left(\frac{(x-1)^2}{2}\right), \quad x \in \mathbb{R}, \quad P(1)=1,
$$
Determine the polynomials $P(x)$ and $Q(x)$.