From CPhill answer :
x^2 + ( 2 + k)x + 28
the roots are the same R1 = R2
From Vieta
R1R1 = 28 so R1 = +- sqrt 28
and - ( R1 + R1) /a = -R1 - R1 = 2 + k (a = 1 in this case)
- 2R1 = 2+k
- 2(+- sqrt28) - 2 = k
- 4 sqrt7 -2 = k and 4 sqrt 28 -2 = k summed = -4
(as Chris found via different method)