<Question>
Find the remainder when $(5x + 9)^{611} + (x + 5)^{11} + (x - 1)^{11} + 3x^2 + 1$ is divided by $x + 2.$
</Question>

<think>
<Plan_1>
Okay, so I need to find the remainder when the polynomial (5x + 9)^611 + (x + 5)^11 + (x - 1)^11 + 3x^2 + 1 is divided by x + 2. Hmm, let me think about how to approach this.

<prompt_1.1> Recall and describe the Remainder Theorem and how it can be applied to find the remainder of a polynomial when divided by a linear polynomial. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, I remember that when a polynomial is divided by a linear divisor like x + a, the remainder is just the value of the polynomial evaluated at x = -a. This is known as the Remainder Theorem. So in this case, since we're dividing by x + 2, which is the same as x - (-2), the remainder should be the value of the polynomial when x is substituted with -2. That seems straightforward. So instead of having to perform polynomial long division, which would be really tedious given the high exponent 611, I can just plug in x = -2 into the entire expression and compute the result. That will give me the remainder.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Let me write down the polynomial again and substitute x = -2 into each term:

<prompt_2.1> Substitute x = -2 into each term of the polynomial and calculate the result. </prompt_2.1>

Let me check each term again to make sure I didn't make a mistake.

<prompt_2.2> Re-evaluate each term of the polynomial after substituting x = -2 to verify the calculations. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
1. (5x + 9)^611 becomes (5*(-2) + 9)^611
2. (x + 5)^11 becomes (-2 + 5)^11
3. (x - 1)^11 becomes (-2 - 1)^11
4. 3x^2 becomes 3*(-2)^2
5. The constant term is +1.

Let me compute each term step by step.

Starting with the first term: (5*(-2) + 9)^611. Calculating inside the parentheses first: 5*(-2) is -10, so -10 + 9 is -1. So this term simplifies to (-1)^611. Since 611 is an odd number, (-1)^611 is -1.

Next term: (-2 + 5)^11. That's 3^11. Let me compute 3^11. 3^1 is 3, 3^2 is 9, 3^3 is 27, 3^4 is 81, 3^5 is 243, 3^6 is 729, 3^7 is 2187, 3^8 is 6561, 3^9 is 19683, 3^10 is 59049, and 3^11 is 177147. So that term is 177147.

Third term: (-2 - 1)^11. That's (-3)^11. Again, since the exponent is odd, this will be negative. 3^11 is 177147, so (-3)^11 is -177147.

Fourth term: 3*(-2)^2. First, (-2)^2 is 4, so 3*4 is 12.

Fifth term is just +1.
</execution_2.1>

<execution_2.2>
First term: (5x + 9)^611. Substitute x = -2: 5*(-2) + 9 = -10 + 9 = -1. So (-1)^611. Since 611 is odd, this is -1.

Second term: (x + 5)^11. Substitute x = -2: (-2) + 5 = 3. So 3^11. Let me compute that. 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. So that's 177147.

Third term: (x - 1)^11. Substitute x = -2: (-2) - 1 = -3. So (-3)^11. Since the exponent is odd, this is -3^11. Again, 3^11 is 177147, so this term is -177147.

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

Fifth term: 1. That's just 1.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2:
Now let's add all these up:

First term: -1

<prompt_3.1> Sum up all the calculated terms to find the remainder. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Second term: +177147

Third term: -177147

Fourth term: +12

Fifth term: +1

So adding them together:

Start with -1 + 177147 = 177146

Then 177146 - 177147 = -1

Then -1 + 12 = 11

Then 11 + 1 = 12

Wait, so the total sum is 12? Let me verify that again step by step.

First term: -1

Second term: +177147. So -1 + 177147 = 177146

Third term: -177147. So 177146 - 177147 = -1

Fourth term: +12. So -1 + 12 = 11

Fifth term: +1. So 11 + 1 = 12.

So the remainder is 12?
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Hmm, that seems surprisingly simple. Let me check if I missed any terms or made any calculation errors.

<prompt_4.1> Verify the calculations for each term and confirm the sum of the terms. </prompt_4.1>
-----

Wait, but let me check if there's any mistake in substituting x = -2 into each term. Let's verify each substitution again:

<prompt_4.2> Re-verify the substitution of x = -2 into each term and confirm the calculations. </prompt_4.2>
-----

But let me think again: when you have (x - a)^n, the remainder when divided by x + a is (x - a)^n evaluated at x = -a?

<prompt_4.3> Re-evaluate the approach using the Remainder Theorem and confirm the calculations for each term. </prompt_4.3>

But wait, just to be absolutely thorough, let me check once more each term:

<prompt_4.4> Perform a final re-evaluation of each term and confirm the calculations. </prompt_4.4>

However, just to make sure, maybe there's a different approach. For example, perhaps expanding each term modulo (x + 2). But since we are evaluating at x = -2, that's equivalent to substituting x = -2.
So regardless, the result should be the same. Therefore, I think the answer is indeed 12. Let me check once more the addition:

<prompt_4.5> Perform a final verification of the addition of terms and confirm the remainder. </prompt_4.5>
</Plan_4>

<Execution_4>
<execution_4.1>
Let's re-express the terms:

-1 (from first term) + 177147 (second term) = 177146

177146 - 177147 (third term) = -1

-1 + 12 (fourth term) = 11

11 + 1 (fifth term) = 12.
</execution_4.1>

<execution_4.2>
1. (5x + 9)^611 at x = -2: 5*(-2) = -10, -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*(4) = 12. Correct.

5. The constant term is 1. Correct.

Adding them: -1 + 177147 -177147 +12 +1. The 177147 and -177147 cancel each other out, leaving -1 +12 +1 = 12.
</execution_4.2>

<execution_4.3>
Wait, no. The Remainder Theorem says that the remainder of a polynomial f(x) divided by (x - a) is f(a). But here, we are dividing by x + 2, which is (x - (-2)), so a = -2. Therefore, the remainder is f(-2). So yes, substituting x = -2 into the entire polynomial is correct.

Therefore, each term is evaluated at x = -2, and then summed. The calculations seem correct.

But let me check the exponents again. For the first term, (5x +9)^611. At x = -2, 5*(-2) +9 = -10 +9 = -1. (-1)^611 is -1. Correct.

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

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

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

Fifth term: 1. So, yes, adding them all: -1 +177147 -177147 +12 +1. The 177147 and -177147 cancel each other out, leaving -1 +12 +1 =12. So yes, the remainder is 12.
</execution_4.3>

<execution_4.4>
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*(4) = 12. Correct.

5. +1. Correct.

Adding them: -1 +177147 = 177146; 177146 -177147 = -1; -1 +12 = 11; 11 +1 = 12.
</execution_4.4>

<execution_4.5>
First term: -1

Second term: +177147 → Total: 177146

Third term: -177147 → 177146 - 177147 = -1

Fourth term: +12 → -1 +12 = 11

Fifth term: +1 → 11 +1 =12.

Yes, that adds up correctly. Therefore, the remainder is 12.
</execution_4.5>
</Execution_4>

<Plan_5>
Based on execution_4.1:
But let me check if there's any mistake in the arithmetic here. Let's compute the sum step by step again:

<prompt_5.1> Re-calculate the sum of the terms step by step to ensure accuracy. </prompt_5.1>
-----

Based on execution_4.5:
So I think that's the answer.
The remainder is 12.

**Final Answer**
The remainder is \boxed{12}.
</Plan_5>

<Execution_5>
<execution_5.1>
First term: -1

Second term: 177147. So adding these: (-1) + 177147 = 177146

Third term: -177147. So 177146 - 177147 = -1

Fourth term: 12. So -1 + 12 = 11

Fifth term: +1. 11 + 1 = 12.
</execution_5.1>
</Execution_5>

<Final_answer>
Therefore, the remainder when the given polynomial is divided by $x+2$ is $\boxed{12}$.
</Final_answer>
</think>