There are only 9 possibilities: 1111, ..., 9999.
If you add 1 to the polynomial, the result must be divisible by 12.
Such a number must be 1112, 2223, ..., 8889, 10000
To be divisible by 12, it must be divisible by both 3 and 4.
For the sum of digits to be 0 mod 3, the 4th digit of the incremented number must be 0 mod 3.
Candidates so far: 2223, 5556, 8889.
But only 5556 is also divisible by 4.
So, 12n^2 + 12n + 11 = 5555
12n^2 + 12n - 5544 = 0.
A quick check shows that n=21 works, but our friend the quadratic formula gives both solutions:
Solution: n can be either 21 or -22
analytic solution
