Okay, so I need to find the product CD where C and D are integers that satisfy the equation:

\[\frac{C}{x-3} + \frac{D}{x+8} = \frac{4x - 23}{x^2 + 5x - 24}\]

for all real x except -8 and 3. Hmm, this looks like a partial fractions problem. Let me recall how partial fractions work. If we have a rational function, we can express it as a sum of simpler fractions. The denominator here is x² + 5x - 24. Let me factor that to confirm if it's (x - 3)(x + 8). Let's check:

Multiply (x - 3)(x + 8) = x² + 8x - 3x -24 = x² +5x -24. Yes, that's correct. So the denominator factors into (x - 3)(x + 8). Therefore, the original equation is set up for partial fractions where C and D are the numerators over the linear factors.

So the equation is already in the form of partial fractions. To find C and D, we can combine the left-hand side over a common denominator and set it equal to the right-hand side. Then, equate the numerators.

Let me write that out step by step.

The left-hand side (LHS) is:

\[\frac{C}{x - 3} + \frac{D}{x + 8}\]

To combine these into a single fraction, the common denominator is (x - 3)(x + 8). So:

\[\frac{C(x + 8) + D(x - 3)}{(x - 3)(x + 8)} = \frac{4x - 23}{(x - 3)(x + 8)}\]

Since the denominators are equal, the numerators must be equal:

C(x + 8) + D(x - 3) = 4x - 23

Now, expand the left-hand side:

Cx + 8C + Dx - 3D = 4x - 23

Combine like terms:

(C + D)x + (8C - 3D) = 4x - 23

Now, since this equation must hold for all x (except x = 3 and x = -8), the coefficients of corresponding terms must be equal. Therefore, we can set up a system of equations:

1. Coefficient of x: C + D = 4
2. Constant term: 8C - 3D = -23

So now, I need to solve this system of equations for C and D. Let me write them down:

Equation (1): C + D = 4

Equation (2): 8C - 3D = -23

Let me solve equation (1) for one variable, say C = 4 - D, and substitute into equation (2):

8(4 - D) - 3D = -23

Compute 8*4 = 32, so:

32 - 8D - 3D = -23

Combine like terms:

32 - 11D = -23

Subtract 32 from both sides:

-11D = -23 - 32

-11D = -55

Divide both sides by -11:

D = (-55)/(-11) = 5

So D = 5. Then, from equation (1), C = 4 - D = 4 - 5 = -1.

Therefore, C = -1 and D = 5. Then, the product CD is (-1)(5) = -5.

Wait, let me verify this. Substitute C and D back into the original equation to check.

Left-hand side:

(-1)/(x - 3) + 5/(x + 8) = [ -1*(x + 8) + 5*(x - 3) ] / (x - 3)(x + 8)

Compute numerator:

-1*(x + 8) + 5*(x - 3) = -x -8 +5x -15 = ( -x +5x ) + ( -8 -15 ) = 4x -23

Which matches the numerator on the right-hand side. So the solution is correct.

Therefore, CD = (-1)(5) = -5. So the answer should be \boxed{-5}.

But wait, let me double-check my steps again to make sure I didn't make any mistakes.

Starting from the equation:

C(x + 8) + D(x - 3) = 4x -23

Expanding:

Cx + 8C + Dx -3D = 4x -23

Group terms:

(C + D)x + (8C -3D) = 4x -23

Set coefficients equal:

C + D = 4

8C -3D = -23

Solving:

From first equation: C = 4 - D

Substitute into second equation:

8(4 - D) -3D = -23

32 -8D -3D = -23

32 -11D = -23

-11D = -55

D = 5

C = 4 -5 = -1

Yes, that's correct. Then CD = (-1)(5) = -5. The verification step also checks out. So I think this is the right answer.

**Final Answer**
\boxed{-5}
Therefore, the product $CD$ is $\boxed{-5}$.
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