Let \(S\) be the set of positive integral values of \(a\) for which \(\frac{a x^2+2(a+1) x+9 a+4}{x^2-8 x+32}<0, \for…

Given \(\frac{a x^2+2(a+1) x+9 a+4}{x^2-8 x+32}<0 \quad \forall x \in \mathbb{R}\)
In \(x^2-8 x+32\), we have \(D=64-128<0\)

\( \therefore x^2-8 x+32>0\) \(\forall x \in \mathbb{R} \)
\( \Rightarrow a x^2+2(a+1) x+9 a+4<0\) \(\forall x \in R \)
\( \Rightarrow a<0\) and \(D<0\)

Since Question has asked for positive integral values of \(a\)

\( \therefore|S|=0\)

but we need positive integral value of a.
So, No solution