Question 125
Find all polynomials $P(x)$ with real coefficients, such that for every distinct natural $x, y, z, t$ satisfying $x^2 + y^2 + z^2 = 2t^2$ and $\text{GCD}\{x, y, z, t\} = 1$, the following equality holds:
$$
2P^2(t) + 2P(xy + yz + zx) = P^2(x + y + z).
$$
$$
2P^2(t) + 2P(xy + yz + zx) = P^2(x + y + z).
$$