Okay, let's see. The problem is to find the product of all possible values of x in the equation |x - 4| - 10 = 2. Hmm, absolute value equations. Right, absolute value equations usually have two solutions because the expression inside can be either positive or negative. So, I need to solve for x and then multiply the solutions together. Let me think through this step by step.

First, let me rewrite the equation to make it a bit clearer. The equation is |x - 4| - 10 = 2. Maybe I should isolate the absolute value term first. So, if I add 10 to both sides, that should help. Let me do that:

|x - 4| - 10 + 10 = 2 + 10  
Which simplifies to |x - 4| = 12.

Okay, so now we have |x - 4| = 12. The absolute value of something equals 12. That means the expression inside the absolute value, which is (x - 4), can be either 12 or -12. So, I need to solve both cases.

Case 1: x - 4 = 12  
Solving for x here, add 4 to both sides:  
x = 12 + 4  
x = 16.

Case 2: x - 4 = -12  
Again, solving for x, add 4 to both sides:  
x = -12 + 4  
x = -8.

So, the two solutions are x = 16 and x = -8. Now, the question asks for the product of all possible values of x. That means I need to multiply these two solutions together.

Calculating the product: 16 * (-8) = -128.

Wait, let me check if I did that right. 16 times 8 is 128, and since one of them is negative, the product should be negative. So yes, -128. Hmm, that seems straightforward. Let me just verify my steps again.

Starting with |x - 4| -10 = 2. Add 10 to both sides: |x - 4| = 12. Then split into two equations: x - 4 = 12 and x - 4 = -12. Solving both gives x = 16 and x = -8. Multiplying them: 16 * (-8) = -128. Yep, that checks out.

I don't think I made any mistakes here. The key steps were isolating the absolute value, solving both cases, and then multiplying the results. Since the equation was linear after isolating the absolute value, it was straightforward. Maybe if there were more complex equations, but here it's simple. So, the product is -128.
Therefore, the product of all possible values of \( x \) is \boxed{-128}.
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