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.
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.