Okay, so I need to find the remainder when this polynomial is divided by x + 2. Let me think about how to approach this. Hmm, I remember something called the Remainder Theorem. Is that right? The theorem states that if a polynomial f(x) is divided by (x - a), the remainder is f(a). Wait, but here we're dividing by x + 2, which is the same as (x - (-2)). So according to the Remainder Theorem, the remainder should be f(-2). That sounds manageable. So, instead of dealing with the whole polynomial division, I can just substitute x = -2 into the polynomial and compute the value. That should give me the remainder. Let me check if this makes sense.

Let me write down the polynomial: (5x + 9)^611 + (x + 5)^11 + (x - 1)^11 + 3x^2 + 1. So, substituting x = -2 into each term should give me the remainder. Let's do that step by step.

First term: (5*(-2) + 9)^611. Let's compute inside the brackets first: 5*(-2) is -10, so -10 + 9 is -1. Then, (-1)^611. Since 611 is an odd exponent, that's just -1.

Second term: (-2 + 5)^11. That's 3^11. Hmm, 3^11 is a big number. Wait, maybe I can compute that later. Let me note that 3^11 is 177147. Wait, is that right? Let me verify. 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243, 3^6 = 729, 3^7 = 2187, 3^8 = 6561, 3^9 = 19683, 3^10 = 59049, 3^11 = 177147. Yeah, that's correct.

Third term: (-2 - 1)^11. That's (-3)^11. Again, exponent is odd, so this is -3^11. Which is -177147.

Fourth term: 3*(-2)^2. Let's compute (-2)^2 first, which is 4. Then multiply by 3: 3*4 = 12.

Fifth term: 1. That's just 1.

Now, adding all these together:

First term: -1

Second term: +177147

Third term: -177147

Fourth term: +12

Fifth term: +1

Let me add them step by step.

Start with -1 + 177147. That's 177146.

Then add -177147: 177146 - 177147 = -1.

Then add 12: -1 + 12 = 11.

Then add 1: 11 + 1 = 12.

So the total remainder is 12. Wait, that seems straightforward. Let me check again to make sure I didn't make a mistake in any of the steps.

First term: 5*(-2) + 9 = -10 + 9 = -1. (-1)^611 is -1. Correct.

Second term: (-2 + 5) = 3. 3^11 is 177147. Correct.

Third term: (-2 -1) = -3. (-3)^11 is -177147. Correct.

Fourth term: 3*(-2)^2 = 3*4 = 12. Correct.

Fifth term: 1. Correct.

Adding them up: -1 + 177147 = 177146. Then 177146 - 177147 = -1. Then -1 +12 =11. Then 11 +1=12. Yes, that seems right. So the remainder is 12. Hmm, that was easier than I thought. Maybe I was overcomplicating it at first. Let me see if there's another way to approach this, just to confirm.

Alternatively, could I factor out something or use polynomial properties? But since the problem is to divide by x + 2, the Remainder Theorem is the most straightforward method here. I don't see any need for long division or synthetic division here since substituting x = -2 gives the remainder directly. The key here is correctly evaluating each term at x = -2. Let me check each term again:

1. (5x +9)^611 at x = -2: 5*(-2) +9 = -10 +9 = -1. (-1)^611 = -1. Correct.

2. (x +5)^11 at x = -2: (-2 +5) = 3. 3^11 = 177147. Correct.

3. (x -1)^11 at x = -2: (-2 -1) = -3. (-3)^11 = -177147. Correct.

4. 3x^2 at x = -2: 3*(-2)^2 = 3*4 =12. Correct.

5. 1. Correct.

Adding them: (-1) +177147 + (-177147) +12 +1. The 177147 and -177147 cancel each other, leaving -1 +12 +1 =12. Yep, same result. So seems consistent. Therefore, the remainder is 12.

I think that's solid. Let me just think if there's any possible mistake here. For example, did I compute 3^11 correctly? Let me check 3^11 again. 3^1=3, 3^2=9, 3^3=27, 3^4=81, 3^5=243, 3^6=729, 3^7=2187, 3^8=6561, 3^9=19683, 3^10=59049, 3^11=177147. Yes, that's right. Similarly, (-3)^11 is -177147. Correct. So all exponents are computed correctly. The arithmetic adds up: -1 +177147 is 177146, minus 177147 is -1, plus12 is11, plus1 is12. All steps check out. Therefore, the remainder is indeed 12. I think that's the answer.

**Final Answer**
The remainder is \boxed{12}.
Therefore, the remainder when the given polynomial is divided by $x + 2$ is $\boxed{12}$.
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