<Question>
The expression $10x^2-x-24$ can be written as $(Ax-8)(Bx+3),$ where $A$ and $B$ are integers. What is $AB + B$?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out how to factor the quadratic expression 10x² - x - 24 into the form (Ax - 8)(Bx + 3), where A and B are integers. Then, once I find A and B, I have to compute AB + B. Alright, let's start by recalling how to factor quadratics.

The general form of a quadratic is ax² + bx + c. In this case, the given quadratic is 10x² - x - 24. The problem wants me to write this as (Ax - 8)(Bx + 3). So, let's expand (Ax - 8)(Bx + 3) and then compare coefficients with the original quadratic. That should give me equations to solve for A and B.

Expanding (Ax - 8)(Bx + 3):

<prompt_1.1> Expand the expression (Ax - 8)(Bx + 3) and compare it to the original quadratic 10x² - x - 24. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, multiply the terms: Ax * Bx = ABx², then Ax * 3 = 3Ax, -8 * Bx = -8Bx, and -8 * 3 = -24. So combining like terms:

ABx² + (3A - 8B)x - 24.

Now, this should equal the original quadratic 10x² - x - 24. Therefore, the coefficients of corresponding terms must be equal.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
So, let's set up the equations:

1. Coefficient of x²: AB = 10
2. Coefficient of x: 3A - 8B = -1
3. Constant term: -24 (which matches both sides, so no issue there)

So, the key equations are AB = 10 and 3A - 8B = -1. Since A and B are integers, I need to find integer pairs (A, B) such that their product is 10, and 3A - 8B equals -1.

First, let's list all pairs of integers (A, B) that multiply to 10. Remember that both positive and negative pairs are possible because the product is positive.

Possible pairs for (A, B):

<prompt_2.1> List all integer pairs (A, B) that multiply to 10. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
1. (1, 10)
2. (2, 5)
3. (5, 2)
4. (10, 1)
5. (-1, -10)
6. (-2, -5)
7. (-5, -2)
8. (-10, -1)

These are all the possible pairs. Now, for each of these pairs, we can check whether 3A - 8B equals -1.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Let's go through them one by one.

<prompt_3.1> Check if the pair (1, 10) satisfies the equation 3A - 8B = -1. </prompt_3.1>

Next pair: (2,5).

<prompt_3.2> Check if the pair (2, 5) satisfies the equation 3A - 8B = -1. </prompt_3.2>

Next pair: (5,2).

<prompt_3.3> Check if the pair (5, 2) satisfies the equation 3A - 8B = -1. </prompt_3.3>

Next pair: (10,1).

<prompt_3.4> Check if the pair (10, 1) satisfies the equation 3A - 8B = -1. </prompt_3.4>

Now let's check the negative pairs.

Pair (-1, -10):

<prompt_3.5> Check if the pair (-1, -10) satisfies the equation 3A - 8B = -1. </prompt_3.5>

Pair (-2, -5):

<prompt_3.6> Check if the pair (-2, -5) satisfies the equation 3A - 8B = -1. </prompt_3.6>

Pair (-5, -2):

<prompt_3.7> Check if the pair (-5, -2) satisfies the equation 3A - 8B = -1. </prompt_3.7>

Pair (-10, -1):

<prompt_3.8> Check if the pair (-10, -1) satisfies the equation 3A - 8B = -1. </prompt_3.8>
</Plan_3>

<Execution_3>
<execution_3.1>
1. (1, 10): Compute 3(1) - 8(10) = 3 - 80 = -77. That's not equal to -1.
</execution_3.1>

<execution_3.2>
Compute 3*2 - 8*5 = 6 - 40 = -34. That's not -1.
</execution_3.2>

<execution_3.3>
Third pair: (5, 2). Compute 3*5 - 8*2 = 15 - 16 = -1. Oh! That works. So A=5, B=2. Let's check the fourth pair just in case, but maybe it's already found.
</execution_3.3>

<execution_3.4>
3A -8B = 3*10 -8*1 = 30 -8 = 22 ≠ -1. Not good.

Next pair: (1,10). Let's check:

3*1 -8*10 = 3 - 80 = -77 ≠ -1. Nope.

Next pair: (-1, -10):

3*(-1) -8*(-10) = -3 + 80 = 77 ≠ -1. Not.

Next pair: (-2, -5):

3*(-2) -8*(-5) = -6 + 40 = 34 ≠ -1. Nope.

Next pair: (-5, -2):

3*(-5) -8*(-2) = -15 + 16 = 1. Hmm, close but not -1.

Last pair: (-10, -1):

3*(-10) -8*(-1) = -30 +8 = -22 ≠ -1.

So, none of the negative pairs worked. Wait, but that can't be right. Did I make a mistake here? Let me double-check.
</execution_3.4>

<execution_3.5>
Compute 3*(-1) -8*(-10) = -3 + 80 = 77. Not equal to -1. Discard.

Pair (-2, -5):
3*(-2) -8*(-5) = -6 + 40 = 34. Not -1. Discard.

Pair (-5, -2):
3*(-5) -8*(-2) = -15 + 16 = 1. Not -1. Close, but no. Discard.

Pair (-10, -1):
3*(-10) -8*(-1) = -30 + 8 = -22. Not -1. Nope.
</execution_3.5>

<execution_3.6>
3*(-2) -8*(-5) = -6 +40 = 34 ≠ -1. Not good.

Next pair (-5, -2):

3*(-5) -8*(-2) = -15 +16 = 1 ≠ -1.

Next pair (-10, -1):

3*(-10) -8*(-1) = -30 +8 = -22 ≠ -1.
</execution_3.6>

<execution_3.7>
Plugging into 3A -8B: 3*(-5) -8*(-2) = -15 +16 = 1. Not equal to -1. 

Next pair (-10, -1):

Plugging into 3A -8B: 3*(-10) -8*(-1) = -30 +8 = -22. Not equal to -1.
</execution_3.7>

<execution_3.8>
Let's check:

3*(-10) -8*(-1) = -30 +8 = -22 ≠ -1. So nope.

Okay, so none of the negative pairs work. So the only pair that worked was (2,5).
</execution_3.8>
</Execution_3>

<Plan_4>
Based on execution_3.3:
Wait, let me double-check:

<prompt_4.1> Verify the solution (A=5, B=2) by expanding (5x - 8)(2x + 3) and checking against the original quadratic. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
If A=5 and B=2, then (5x -8)(2x +3). Let's expand that again:

5x*2x = 10x²,

5x*3 = 15x,

-8*2x = -16x,

-8*3 = -24.

Combine like terms: 15x -16x = -x. So, 10x² -x -24. Perfect, that's exactly the original quadratic. So A=5 and B=2.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
Therefore, AB + B = (5)(2) + 2 = 10 + 2 = 12. So the answer is 12.

<prompt_5.1> Calculate AB + B using the values of A and B found, and confirm the final answer. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
Wait, but let me make sure there isn't another possible pair. Wait, when I checked the negative pairs, did I miscalculate? Let me check pair (-5, -2) again. 

If A=-5 and B=-2, then 3*(-5) -8*(-2) = -15 +16=1. Which is not -1. So that's not. 

Wait, maybe I missed some pairs? Let me think. The factors of 10 are (1,10), (2,5), (5,2), (10,1), and the negatives. I think that's all. So since only (5,2) worked, that must be the answer.
</execution_5.1>
</Execution_5>

<Plan_6>
Based on execution_5.1:
Therefore, the answer is 12. But let me just confirm once again. Let's factor 10x² -x -24 as (5x -8)(2x +3).

<prompt_6.1> Confirm the solution by factoring the quadratic expression again and stating the final answer. </prompt_6.1>
</Plan_6>

<Execution_6>
<execution_6.1>
Expanding:

First term: 5x*2x =10x²,

Outer: 5x*3=15x,

Inner: -8*2x= -16x,

Last: -8*3= -24.

Combine: 10x² + (15x -16x) -24 =10x² -x -24. Correct. So the factorization is correct, so A=5, B=2. Then AB + B =5*2 +2=10 +2=12. So 12 is the answer.

But wait, the problem says "where A and B are integers". So maybe there is another factorization where A and B are different? Let me think. For example, maybe A and B are fractions, but the problem states they are integers, so no. Since we checked all possible integer pairs, and only (5,2) works. Therefore, 12 is the correct answer.
</execution_6.1>
</Execution_6>

<Final_answer>
Therefore, the final answer is $\boxed{12}$.
</Final_answer>
</think>