$\text{ The number of positive integral solutions of }{x}^2+9<(x+3{)}^2<8x+25\text{ is }$

$\text{ We have }{x}^2+9<(x+3{)}^2\text{ and }(x+3{)}^2<8x+25$

$\begin{array}{l}\Rightarrow 6x>0\text{ and }{x}^2-2x-16<0\\ \Rightarrow x>0\text{ and }(x-1{)}^2<17\\ \Rightarrow x>0\text{ and }-\sqrt{17}+1<x<1+\sqrt{17}\\ \Rightarrow 0<x<1+\sqrt{17}\\ \Rightarrow \text{ integral values are }x=1,2,3,4,5\end{array}$