Let \(S\) be the set of all real numbers \(\alpha\) such that the function \(\frac{x^2+5x+\alpha}{x^2 + 7x - 44}\) can be expressed as a quotient of two linear functions. What is the sum of the elements of \(S\)?
+600 \(x^2+7x-44=(x+11)(x-4)\). So \((x-4)(x+9)=x^2+5x-36\rightarrow a=-36\). And \((x+11)(x-6)=x^2+5x-66\rightarrow a=-66\). So \(\boxed{a=-36,-66}\)
