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

Note that  x^2  + 7x  - 44  can be factored as  ( x + 11) ( x - 4)

So  we need  to  see if  x^2 + 5x + a    can be  factored using one - or both  - of  the factors above

Note  that if   a  = -36   we have  the factrization  ( x + 9) ( x - 4)

And the ratio of linear functions  is   ( x + 9)  / ( x + 11)

And  if a    =    - 66   we have the factorization  ( x + 11) ( x - 6)

And the ratio of linear  functions is  ( x - 6) / ( x - 4)

So  a =  -36   or a =  -66

And their  sum is   -102