discriminant b^2-4ac >0 for two discrete zeroes hence providing 2 factors

144-4 (2) k>0

144-8k>0

8k<144

k<144/8

k<18

2x^2+12x+k, k>0=

2(x^2+6x+9-9)+k=

2(x+3)^2-18+k=

[sqrt(2)(x+3)]^2-[sqrt(18-k)]^2

For real factors,

18>=k>0=>

Take k=10, then

[sqrt(2)(x+3)]^2-[2sqrt(2)]^2=

2(x+3+2)(x+3-2)=

2(x+5)(x+1)

Take k=16, then

[sqrt(2)(x+3)]^2-[sqrt(2)]^2=

2(x+3+1)(x+3-1)=

2(x+4)(x+2)

Take k=18

[sqrt(2)(x+3)]^2-0=

2(x+3)^2

So, k is 10,16 &18.