Nice work, 720Dreaming.
The answer is D.

Here's my full solution as well:
(1) one option is to plug in 2 values (or 3 to provide a little more certainty) for x such that x+1 is divisible by 3 (e.g., x=2, 5, 8 etc) and check whether each value of x yields the same remainder when x^2 - 4x + 11 is divided by 3
(2) Recognize that this information is the same information that was provided in (1). If x+1 is divisible by 3, then it must follow that x-2 is divisible by 3.
When (1) and (2) provide identical information, then the correct answer must be either D or E.
Let's see when the correct answer is D.
When we look at the expression x^2 - 4x + 11, we can see that it’s ALMOST possible to factor out x-2 from this expression. In fact (x-2)^2 = x^2 + 4x + 4
So, we can rewrite x^2 - 4x + 11 as x^2 + 4x + 4 +7, which is the same as (x-2)^2 +7, which is the same as (x-2)^2 +6 + 1
Since x-2 is divisible by 3, it must be true that (x-2)^2 is divisible by 3 and it follows that (x-2)^2 +6 is divisible by 3
So, (x-2)^2 +6 + 1 (aka x^2 - 4x + 11) must have a remainder of 1 when divided by 3