Two consecutive odd integers can be represented by 2x-1 and 2x+1. Their product is 4x^2-1 and their sum is 4x. So, 4x^2-1=3(4x+6)=12x+18. This is the quadratic: 4x^2-12x-19=0. This has no rational solutions.
Or, reading the question slightly differently: 4x^2-1=12x+6, so 4x^2-12x-7=0.
In the second case, we can factorise: (2x-7)(2x+1)=0, so x=3.5 or -0.5. The two integers are: 6 and 8, or -2 and 0. None of these are odd, so it isn't possible to find two consecutive odd integers.
Why did I think there were two interpretations of the question? "3 times the sum of the integers plus 6" is ambiguous: is it 3(sum+6) or 3*sum+6? Judging by the result, it was the latter.