jmc

The left inequality becomes $5x-1<x^2+2x+1,$ or $$x^2-3x+2>0.$$This factors as $(x-1)(x-2)>0,$ and the solution to $x\in (-\infty ,1)\cup (2,\infty ).$

The right inequality becomes $x^2+2x+1<7x-3,$ or $$x^2-5x+4<0.$$This factors as $(x-1)(x-4)<0,$ and the solution is $x\in (1,4).$

The intersection of $(-\infty ,1)\cup (2,\infty )$ and $(1,4)$ is $\boxed{(2,4)}.$