Okay, let's tackle this problem step by step. The question is asking for the value of the expression (3x - 2)(4x + 1) - (3x - 2)4x + 1 when x = 4. Hmm, let me first make sure I understand the expression correctly. It looks like there's a subtraction between two terms: the first term is (3x - 2)(4x + 1), and the second term is (3x - 2)4x. Then, there's an additional +1 at the end. Wait, no, hold on. Let me parse the original expression again. It's written as (3x-2)(4x+1) - (3x-2)4x +1. So, it's [(3x - 2)(4x +1)] minus [(3x -2)4x] plus 1. Got it.

First, maybe I should simplify the expression algebraically before plugging in x = 4. That might make the calculation easier. Let me try expanding the terms. Let's start with the first part: (3x - 2)(4x + 1). To expand this, I'll use the distributive property (FOIL method). 

First, multiply 3x by 4x: 3x * 4x = 12x². Then, 3x * 1 = 3x. Next, -2 * 4x = -8x. Finally, -2 * 1 = -2. So combining those terms: 12x² + 3x - 8x - 2. Combine like terms: 12x² -5x -2. 

Now, the second part is (3x - 2)4x. Let's distribute the 4x across (3x -2). 4x * 3x = 12x², and 4x * (-2) = -8x. So that gives 12x² -8x.

Now, putting it all together: The original expression is (12x² -5x -2) - (12x² -8x) +1. Let's subtract the second term from the first. 

So, distribute the negative sign: 12x² -5x -2 -12x² +8x +1. Now, combine like terms. The 12x² and -12x² cancel each other out. Then, -5x +8x = 3x. Then, -2 +1 = -1. So the simplified expression is 3x -1. 

Wow, that's a big simplification! So instead of dealing with quadratics, the entire expression simplifies to 3x -1. Then, substituting x =4 into this simplified expression: 3*4 -1 = 12 -1 = 11. So the answer should be 11. 

But wait, let me double-check that I didn't make a mistake in simplifying. Let me go through the steps again. 

First part: (3x -2)(4x +1) = 12x² +3x -8x -2 = 12x² -5x -2. Correct. 

Second part: (3x -2)4x = 12x² -8x. Correct. 

Then subtracting the second part from the first: 12x² -5x -2 -12x² +8x +1. The 12x² terms cancel. Then, -5x +8x = 3x. -2 +1 = -1. So indeed, 3x -1. Then, substituting x=4: 3*4=12, 12-1=11. 

Alternatively, maybe I can verify by plugging x=4 into the original expression without simplifying. Let's try that to confirm. 

First, compute (3x -2) when x=4: 3*4=12, 12-2=10. 

Then compute (4x +1) when x=4: 4*4=16, 16+1=17. So the first term is 10*17=170. 

Next, compute (3x -2) again, which is 10. Multiply by 4x: 4*4=16, so 10*16=160. 

Then subtract that from the first term: 170 -160 =10. Then add 1: 10 +1=11. 

Same result. So both methods give 11. Therefore, the answer is 11. 

Alternatively, maybe I can check using another approach, like expanding the original expression step by step without simplifying first. Let's see. 

Original expression: (3x -2)(4x +1) - (3x -2)4x +1. Let's distribute the terms. 

First term: (3x -2)(4x +1) = 12x² -5x -2. 

Second term: -(3x -2)4x = -12x² +8x. 

Then add 1. 

So combining: 12x² -5x -2 -12x² +8x +1. Again, same as before: 3x -1. 

Plugging in x=4: 11. 

So all methods lead to the same answer. Therefore, I'm confident the answer is 11.
Final Answer
The value of the expression when \( x = 4 \) is \boxed{11}.
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