VN-TST 1994 Problem 3

Let $P(x)$ be given a polynomial of degree 4, having 4 positive roots. Prove that the equation $$(1-4 \cdot x) \cdot \frac{P(x)}{x^2} + (x^2 + 4 \cdot x - 1) \cdot \frac{P'(x)}{x^2} - P''(x) = 0$$ has also 4 positive roots.