SCHWEITZER 1955 Problem 7

Prove that for any odd prime number $p$ , the polynomial $2(1+x^{ \frac{p+1}{2} }+(1-x)^{\frac {p+1}{2}})$ is congruent mod $p$ to the square of a polynomial with integer coefficients. (N. 21)

*This problem was proposed by P. Erdõs in the American Mathematical Monthly 53 (1946), p. 594