The number of positive integral solutions of $x^2+9<(x+3)^2 < 8x + 25$, is _____.

5

We have, $x^2+9<(x+3)^2 < 8x + 25$ $⇒x^2+9<x^2+6x+9 <8x + 25$ $⇒x^2+9<x^2+6x+9$ and $x^2+6x+9 <8x+25$ $⇒6x>0$ and $x^2 -2x-16 <0$ $⇒x>0$ and $1-\sqrt{17} <x<1+ \sqrt{17}$ $⇒0<x<1+ \sqrt{17}⇒x=1,2,3,4,5$