<Question>
Simplify $4(3r^3+5r-6)-6(2r^3-r^2+4r)$, and express your answer in the form $Ar^2 + Br + C$, where $A$, $B$, and $C$ are integers.
</Question>

<think>
<Plan_1>
Okay, let me try to simplify this expression step by step. The problem is to simplify 4 times the quantity (3r³ + 5r - 6) minus 6 times the quantity (2r³ - r² + 4r), and then express the answer in the form Ar² + Br + C. Alright, so first, I need to distribute the 4 and the -6 into each of the respective parentheses. Let me write that out.

Starting with the first term: 4 multiplied by each term inside the first parentheses. So, 4 times 3r³ is 12r³. Then, 4 times 5r is 20r. Next, 4 times -6 is -24. So that part becomes 12r³ + 20r - 24.

Now the second term: -6 multiplied by each term in the second parentheses. Let's do that step by step.

<prompt_1.1> Distribute -6 across the terms in the second parentheses and combine like terms with the first part. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, -6 times 2r³ is -12r³. Then, -6 times -r² would be +6r² because a negative times a negative is positive. Next, -6 times 4r is -24r. So the second part becomes -12r³ + 6r² - 24r.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, I need to combine the two parts together. Let me write them next to each other:

<prompt_2.1> Combine the two parts obtained from distributing 4 and -6, and arrange the terms by the power of r. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
12r³ + 20r - 24 -12r³ + 6r² - 24r.

Wait, I need to make sure to include the signs correctly. The first part is 12r³ + 20r -24, and then subtracting the second part which is -12r³ +6r² -24r. Wait, actually, no. Wait, the original expression is 4*(... ) -6*(... ), so when we distribute, the second term is subtracted as a whole. So when combining, it's 12r³ +20r -24 -12r³ +6r² -24r. Let me confirm that. Yes, because subtracting each term in the second parentheses: -6*2r³ is -12r³, -6*(-r²) is +6r², -6*4r is -24r. So yes, that's correct. Then the constants are just -24.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Now, let's combine like terms. Let's look for the same powers of r.

<prompt_3.1> Combine like terms for the r³, r², r, and constant terms separately. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Starting with r³ terms: 12r³ -12r³. Those cancel each other out, right? 12 -12 is 0, so there's no r³ term in the final expression.

Next, the r² term: there's only one, which is +6r² from the second part. So that's the r² term.

Then, the r terms: 20r -24r. That should be (20 -24)r, which is -4r.

Finally, the constant term: -24. There are no other constants to combine with, so that remains.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Putting it all together, we have 6r² -4r -24.

Wait a second, the problem says to express the answer in the form Ar² + Br + C. So that's exactly the form we have here. Let me check if all the terms are accounted for.

<prompt_4.1> Verify the simplified expression and ensure all terms are correctly combined. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
The original expression after distribution had r³, r², r, and constant terms. The r³ terms canceled out, so the highest power is r². So the simplified form is 6r² -4r -24. Therefore, A is 6, B is -4, and C is -24. So the answer should be 6r² -4r -24.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
Let me just double-check my distribution and combining steps to make sure I didn't make any mistakes.

<prompt_5.1> Double-check the distribution and combining steps to ensure no mistakes were made. </prompt_5.1>

Hmm, but let me check once more for any possible errors. Maybe I missed a sign or something.

<prompt_5.2> Re-evaluate the distribution and combining steps to check for any possible errors. </prompt_5.2>

Wait, but let me check if there's a possibility that the problem expects the terms ordered differently, but the question specifies Ar² + Br + C, so our answer is in the correct order.

<prompt_5.3> Confirm the order of terms and ensure the answer matches the required form. </prompt_5.3>

I think that's it. Let me just verify once again.

<prompt_5.4> Perform a final verification of the solution to ensure all terms are correctly combined and simplified. </prompt_5.4>
</Plan_5>

<Execution_5>
<execution_5.1>
First distribution: 4*(3r³ +5r -6) = 12r³ +20r -24. That seems right. Then, -6*(2r³ -r² +4r) = -12r³ +6r² -24r. Yes, because -6 times -r² is +6r². Then combining:

12r³ -12r³ = 0.

6r² remains.

20r -24r = -4r.

-24 remains.
</execution_5.1>

<execution_5.2>
First distribution: 4*(3r³ +5r -6) = 12r³ +20r -24. That seems correct. Then, -6*(2r³ -r² +4r) = -12r³ +6r² -24r. Yes, that's right. Then combining:

12r³ -12r³ = 0.

6r² remains.

20r -24r = -4r.

-24 remains. So yes, the final expression is 6r² -4r -24.
</execution_5.2>

<execution_5.3>
So 6r² -4r -24 is correct. Alright, I think that's the final answer.
</execution_5.3>

<execution_5.4>
Original expression:

4(3r³ +5r -6) -6(2r³ -r² +4r)

After distribution:

12r³ +20r -24 -12r³ +6r² -24r

Combine like terms:

12r³ -12r³ = 0

6r² remains

20r -24r = -4r

-24 remains

So yes, 6r² -4r -24 is correct. Therefore, the answer should be 6r² -4r -24.
</execution_5.4>
</Execution_5>

<Final_answer>
Therefore, the simplified expression is $\boxed{6r^2-4r-24}$.
</Final_answer>
</think>