<Question>
What is the value of $$
(3x-2)(4x+1)-(3x-2)4x+1
$$ when $x=4$?
</Question>

<think>
<Plan_1>
Okay, let's see. I need to find the value of the expression (3x - 2)(4x + 1) - (3x - 2)4x + 1 when x is 4. Hmm, first, I should probably simplify the expression before plugging in x=4. That might make the calculation easier. Let me look at the expression again: (3x - 2)(4x + 1) minus (3x - 2)4x plus 1. Wait, so there are two terms being subtracted and then adding 1. Let me write it out step by step to make sure I don't misplace any terms.

First, expand (3x - 2)(4x + 1). To do that, I can use the distributive property, also known as FOIL (First, Outer, Inner, Last). Let's compute each part:

<prompt_1.1> Expand the expression (3x - 2)(4x + 1) using the distributive property. </prompt_1.1>

Now, the second part is -(3x - 2)4x. Let me handle that. First, distribute the 4x into (3x - 2):

<prompt_1.2> Expand the expression -(3x - 2)4x and simplify. </prompt_1.2>
-----

Let me check if there's another way to approach this, maybe by factoring out common terms first. Let's look at the original expression: (3x - 2)(4x + 1) - (3x - 2)4x + 1. Notice that both the first and second terms have a common factor of (3x - 2). So, maybe factor that out:

<prompt_1.3> Factor out the common term (3x - 2) from the expression and simplify. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
First terms: 3x * 4x = 12x²
Outer terms: 3x * 1 = 3x
Inner terms: -2 * 4x = -8x
Last terms: -2 * 1 = -2

Combine those: 12x² + 3x - 8x - 2. Then combine like terms: 12x² - 5x - 2.
</execution_1.1>

<execution_1.2>
First, multiply 4x by 3x: 12x². Then, 4x multiplied by -2: -8x. So, that term becomes -12x² + 8x. Wait, no, actually, the entire (3x - 2)4x is being subtracted. So, actually, the expression is - (3x - 2) * 4x. So, first compute (3x - 2)4x, which is 12x² - 8x, and then the negative sign makes it -12x² + 8x. Okay, that's correct.

So putting it all together, the first part (3x - 2)(4x + 1) expanded is 12x² + 3x - 8x - 2, which simplifies to 12x² -5x -2. Then subtract the second term, which is 12x² -8x, so subtracting that gives:

(12x² -5x -2) -12x² +8x +1. Wait, hold on, the original expression is (3x-2)(4x+1) - (3x-2)4x +1. So after expanding the first two terms, we have 12x² -5x -2 -12x² +8x +1. Let me combine like terms here.

The 12x² and -12x² cancel out. Then, -5x +8x is 3x. Then, -2 +1 is -1. So the entire expression simplifies to 3x -1. Oh! So all that simplifies down to just 3x -1. That's much simpler!
</execution_1.2>

<execution_1.3>
(3x - 2)[(4x + 1) - 4x] + 1.

Let me compute what's inside the brackets: (4x + 1) - 4x. The 4x and -4x cancel each other, leaving just 1. So then the expression becomes (3x - 2)(1) + 1, which simplifies to (3x - 2) + 1. Then, combining like terms: 3x - 2 + 1 = 3x - 1.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2:
Then, substituting x=4 into 3x -1: 3*4 -1 = 12 -1 = 11.
-----

Based on execution_1.3:
Therefore, the answer is 11.

Just to be thorough, let me compute the original expression step by step with x=4 without simplifying first, just to confirm:

<prompt_2.1> Compute the original expression step by step with x=4 without simplifying first. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
First, compute (3x -2)(4x +1) when x=4.

3*4 = 12, 12 -2 =10. So, 3x-2=10.

4*4=16, 16 +1=17. So, 4x +1=17.

Multiply them: 10*17=170.

Next term: (3x -2)4x. We already have 3x-2=10 when x=4. 4x=16. So 10*16=160. But this term is subtracted, so -160.

Then, +1 at the end.

So total expression is 170 -160 +1 = 11.
</execution_2.1>
</Execution_2>

<Final_answer>
Therefore, the value of the given expression when $x=4$ is $\boxed{11}$.
</Final_answer>
</think>