Here's my best attempt....
x^2 + 7x - 44 can be factored as (x + 11) ( x - 4)
So....we are looking for two "a's" such that x^2 + 5x + a has factors of
(x + 11) or ( x - 4)
So
x - 6
x + 11 [ x^2 + 5x + a ]
x^2 + 11x
_____________
-6x + a
-6x - 66
_________
a + 66 must be 0 ....so......a =-66
Proof
x^2 + 5x - 66 (x + 11) ( x - 6) x - 6
_____________ = _____________ = _______
x^2 + 7 x - 44 (x + 11)(x - 4) x - 4
Also
x + 9
x - 4 [ x^2 + 5x + a ]
x^2 - 4x
_____________
9x + a
9x - 36
_______
a+ 36 must = 0 ....so.......a =-36
Second proof
x^2 + 5x - 36 ( x - 4) ( x + 9) x + 9
___________ = _____________ = _______
x^2 + 7x - 44 (x - 4) ( x + 11) x + 11
So either a = -66 or a = - 36
And the sum of the possible a's = - [ 66 + 36 ] = - 102