Okay, so I need to simplify this algebraic expression: 4 times (3r³ + 5r -6) minus 6 times (2r³ - r² +4r). Then express the result in the form Ar² + Br + C. Alright, let's break this down step by step.

First, I remember that when simplifying expressions like this, I need to distribute the coefficients outside the parentheses to each term inside. That means multiplying each term inside the first parentheses by 4 and each term inside the second parentheses by -6. Then, after distributing, I can combine like terms. Let me make sure I do this correctly.

Starting with the first part: 4 times (3r³ +5r -6). Let's distribute the 4 to each term. So, 4*3r³ is 12r³. Then 4*5r is 20r. And 4*(-6) is -24. So that part becomes 12r³ +20r -24. Got that.

Now the second part: -6 times (2r³ - r² +4r). Let's distribute the -6 here. First term: -6*2r³ is -12r³. Second term: -6*(-r²) is +6r². Third term: -6*4r is -24r. So that part becomes -12r³ +6r² -24r.

Now, combining both parts. The entire expression is the first part minus the second part. Wait, actually, the original expression is 4(...) -6(...), so the second part is already multiplied by -6. So when we expanded the second part, we already accounted for the negative sign. So now we have:

First part: 12r³ +20r -24

Second part: -12r³ +6r² -24r

So the whole expression is (12r³ +20r -24) + (-12r³ +6r² -24r). Now, let's combine these two polynomials by adding like terms.

Let's list out all the terms:

From the first part:
12r³
20r
-24

From the second part:
-12r³
6r²
-24r

Now, let's group them by degree:

r³ terms: 12r³ -12r³ = 0r³

r² terms: 6r²

r terms: 20r -24r = -4r

Constants: -24

So putting it all together, the simplified expression is 6r² -4r -24. Let me check that again to make sure I didn't make a mistake.

Wait, let's verify each step again.

First distribution:

4*(3r³) = 12r³

4*5r = 20r

4*(-6) = -24

So first part is correct.

Second distribution:

-6*(2r³) = -12r³

-6*(-r²) = +6r²

-6*(4r) = -24r

So second part is also correct.

Then combining:

12r³ -12r³ = 0, so r³ terms cancel out.

6r² remains.

20r -24r = -4r

-24 remains.

So yes, the simplified form is 6r² -4r -24. So in the form Ar² + Br + C, A is 6, B is -4, and C is -24. Therefore, the answer should be 6r² -4r -24.

Wait, but the problem says to express it in the form Ar² + Br + C. So that's exactly what we have here. Let me just make sure there are no like terms left that I missed.

Looking again: after combining, we have 6r², -4r, and -24. No other terms. So that should be the final answer. I think that's correct. Let me check once more by plugging in a value for r and seeing if both the original and simplified expressions give the same result.

Let's pick r = 1.

Original expression:

4*(3*1³ +5*1 -6) -6*(2*1³ -1² +4*1)

Calculate inside the first parentheses: 3 +5 -6 = 2. Then 4*2 =8.

Inside the second parentheses: 2 -1 +4 =5. Then -6*5 = -30.

So total is 8 -30 = -22.

Simplified expression: 6*1² -4*1 -24 =6 -4 -24 = -22. Same result. Good.

Another test, r=2.

Original:

4*(3*8 +5*2 -6) -6*(2*8 -4 +8)

First parentheses: 24 +10 -6=28. 4*28=112.

Second parentheses:16 -4 +8=20. -6*20=-120.

Total:112 -120= -8.

Simplified:6*4 -4*2 -24=24 -8 -24= -8. Same result. Perfect.

So the simplified expression is correct. Therefore, the answer is 6r² -4r -24.
Therefore, the simplified expression is $\boxed{6r^2 -4r -24}$.
</think>