Question 2
Two real polynomials $f(x),g(x)$ are called proportional if there exists $q\in\mathbb R$ such that $f(x)=qg(x)$ for every real $x$.
1. Prove that there exists a real polynomial $P(x)$ of degree 1999 for which $P(x)^2-4$ and $(P'(x))^2(x^2-4)$ are proportional.
2. How many degree-1999 polynomials satisfy this condition?
1. Prove that there exists a real polynomial $P(x)$ of degree 1999 for which $P(x)^2-4$ and $(P'(x))^2(x^2-4)$ are proportional.
2. How many degree-1999 polynomials satisfy this condition?