$Thenumberof\mathrm{int}egeralvaluesofxsatisfying5x-1<{(x+1)}^2<7x-3is$

$\begin{array}{l}Given5x-1<{(x+1)}^2<7x-3\\ Let5x-1<{(x+1)}^2\\ 5x-1<x^2+1+2x\\ x^2-3x+2>0\\ (x-2)(x-1)>0\\ x>2orx<1\\ Let{(x+1)}^2<7x-3\\ \Rightarrow x^2+1+2x<7x-3\\ \Rightarrow x^2-5x+4<0\\ \Rightarrow (x-1)(x-4)<0\\ 1<x<4\\ \therefore numberofcommonrootsare\text{'}2\text{'}\end{array}$