Putnam 1981 A5

Let $P(x)$ be a polynomial with real coefficients and form the polynomial $$Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ).$$ Given that the equation $P(x) = 0$ has $n$ distinct real roots exceeding $1$ , prove or disprove that the equation $Q(x)=0$ has at least $2n - 1$ distinct real roots.