Question 1
(BUL 2) Consider a sequence of polynomials $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$, where $P_0(x)=2$, $P_1(x)=x$ and for every $n\ge1$ the following equality holds:
$$P_{n+1}(x)+P_{n-1}(x)=xP_n(x).$$
Prove that there exist three real numbers $a,b,c$ such that for all $n\ge1$,
$$(x^2-4)[P_n(x)^2-4]=[aP_{n+1}(x)+bP_n(x)+cP_{n-1}(x)]^2.\tag{1}$$
$$P_{n+1}(x)+P_{n-1}(x)=xP_n(x).$$
Prove that there exist three real numbers $a,b,c$ such that for all $n\ge1$,
$$(x^2-4)[P_n(x)^2-4]=[aP_{n+1}(x)+bP_n(x)+cP_{n-1}(x)]^2.\tag{1}$$