Let $y=\frac{x^{2}+34 x-71}{x^{2}+2 x+7}$
$\Rightarrow \quad x^{2} y+2 x y-7 y=x^{2}+34 x-71$
$\Rightarrow \quad(x)+2 x y-7 y-x^{2}-34 x+71=0$
$(y-1) x^{2}+(2 y-34) x+(71-7 y)=0$
It is in the form of $ax^2+bx+c=0$
Where a = $a=(y-1) \quad b=2 y-34, c=71-7 y$
Since $x$ is real $ \Rightarrow \Delta \geq 0 \Rightarrow b^{2}-4 a c \geq 0$
$\Rightarrow(2 y-34)^{2}-4(y-1)(7-7 y) \geq 0$
$\Rightarrow \quad 4 y^{2}+1156-136 y-4\left(7 y-7 y^{2}-71+7 y\right) \geq 0$
$\Rightarrow 4 y^{2}+1156-136 y-4\left(78 y-7 y^{2}-71\right) \geq 0$
$\Rightarrow 4 y^{2}+1156-136 y-312 y+28 y^{2}+284 \geq 0$
$\Rightarrow \quad 32 y^{2}-448 y+1440 \geq 0$
$\Rightarrow \quad y^{2}-14 y+45 \geq 0$
$\Rightarrow \quad y^{2}-9 y-5 y+45 \geq 0$
$\Rightarrow y(y-9)-5(y-9)>0$
$\Rightarrow(y-9)(y-5) \geq 0 \Rightarrow y \in(-\infty, 5] \cup[9, \infty)$
$\Rightarrow$ y doest not lie tetween 5 and 9
$\Rightarrow$ Given Expresion doesnot lie between 5 and 9
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