jmc

Using the difference of squares factorization, we have $$\begin{array}{rl}15^4+2\times 15^2+1-14^4& =(15^2+1{)}^2-(14^2{)}^2\\ & =(15^2+1-14^2)(15^2+1+14^2)\\ & =(15^2-14^2+1)(422)\\ & =((15-14)(15+14)+1)(2\cdot 211)\\ & =30\cdot 2\cdot 211.\end{array}$$Since $211$ is a prime and is larger than the other factor, we see that $\boxed{211}$ is the largest prime factor.