Write a Polynomial in Terms of Its Roots

(2016 AIME I #11) Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1) = (x+2)P(x)$ for every real $x,$ and $(P(2))^2 = P(3).$ Then $P\left(\frac{7}{2}\right) = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$