Okay, so I need to figure out the values of A and B such that when I multiply (Ax - 8)(Bx + 3), it equals the quadratic expression 10x² - x - 24. Then, once I find A and B, I have to compute AB + B. Let me start by expanding the product of the two binomials and then compare coefficients to solve for A and B.

First, let's expand (Ax - 8)(Bx + 3). Using the distributive property (FOIL), the product would be:

Ax * Bx + Ax * 3 - 8 * Bx - 8 * 3

Simplifying each term:

ABx² + 3Ax - 8Bx - 24

Now, combine like terms:

ABx² + (3A - 8B)x - 24

This should be equal to the original expression 10x² - x - 24. Therefore, the coefficients of corresponding terms must be equal. That gives us a system of equations:

1. Coefficient of x²: AB = 10
2. Coefficient of x: 3A - 8B = -1
3. Constant term: -24 = -24 (which is already satisfied)

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) that satisfy both equations.

Let me list the pairs of integers whose product is 10. The pairs are (1,10), (2,5), (5,2), (10,1), (-1,-10), (-2,-5), (-5,-2), (-10,-1). Now, I need to test these pairs in the second equation 3A - 8B = -1.

Let me start with the positive pairs first.

1. A=1, B=10: 3(1) - 8(10) = 3 - 80 = -77 ≠ -1. Not valid.
2. A=2, B=5: 3(2) - 8(5) = 6 - 40 = -34 ≠ -1. Not valid.
3. A=5, B=2: 3(5) - 8(2) = 15 - 16 = -1. Oh, this works! So A=5, B=2 is a solution.
4. A=10, B=1: 3(10) -8(1)=30 -8=22≠-1. Not valid.

Now checking the negative pairs:

5. A=-1, B=-10: 3(-1) -8(-10)= -3 +80=77≠-1. No.
6. A=-2, B=-5: 3(-2) -8(-5)= -6 +40=34≠-1. No.
7. A=-5, B=-2: 3(-5) -8(-2)= -15 +16=1≠-1. No.
8. A=-10, B=-1: 3(-10) -8(-1)= -30 +8=-22≠-1. No.

So the only pair that works is A=5 and B=2. Therefore, AB is 5*2=10, which matches the first equation. Then AB + B is 10 + 2 = 12. Wait, but let me confirm this again.

Wait, AB is 10, and B is 2, so AB + B is 10 + 2 = 12. That seems straightforward. Let me just verify by plugging A=5 and B=2 back into the original expression.

(5x -8)(2x +3) = 5x*2x +5x*3 -8*2x -8*3 = 10x² +15x -16x -24 = 10x² -x -24. Yes, that's correct. So the coefficients check out. Therefore, AB + B is indeed 12.

But wait, let me make sure there are no other possible solutions. The problem states that A and B are integers, so we considered all possible factor pairs of 10. Since only (5,2) worked in the second equation, that must be the solution. Therefore, the answer is 12.

Just to recap:

1. Expanded the product.
2. Set up equations by comparing coefficients.
3. Solved for A and B by testing integer pairs.
4. Verified the solution by plugging back into the original expression.
5. Calculated AB + B.

I don't see any mistakes in this process. The answer should be 12.
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