Since the square roots of 196 are +14 and -14, 

x2 + kx + 196 = 0

to be a perfect square trinomial, it would have to factor as

(x+14)2 = 0  or (x-14)2 = 0

or

(x+14)(x+14) = 0 or (x-14)(x-14) = 0  

which when multiplied out gives

x2 + 28x + 196 = 0 or x2 - 28x + 196 = 0

So k would have to be +28 or -28

Edwin