If `|x+1|+|x^(2)-3x+2|=|x^(2)-4x+1|`, then sum of the natural number solutions of equation is

Explanation

To solve the equation \( |x+1| + |x^2 - 3x + 2| = |x^2 - 4x + 1| \), we will analyze the expressions inside the absolute values step by step.

### Step 1: Identify the expressions
We have three expressions to consider:
1. \( |x + 1| \)
2. \( |x^2 - 3x + 2| \)
3. \( |x^2 - 4x + 1| \)

### Step 2: Factor the quadratic expressions
Let's factor the quadratic expressions:
- \( x^2 - 3x + 2 = (x - 1)(x - 2) \)
- \( x^2 - 4x + 1 = (x - 2)^2 - 3 = (x - 2 - \sqrt{3})(x - 2 + \sqrt{3}) \)

### Step 3: Find the critical points
The critical points occur where each expression inside the absolute value is zero:
1. From \( x + 1 = 0 \) we get \( x = -1 \).
2. From \( x^2 - 3x + 2 = 0 \) we get \( x = 1 \) and \( x = 2 \).
3. For \( x^2 - 4x + 1 = 0 \), we can find the roots using the quadratic formula:
\[ x = \frac{4 \pm \sqrt{(4)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3} \]
So, the critical points are \( -1, 1, 2, 2 - \sqrt{3}, 2 + \sqrt{3} \).

### Step 4: Test intervals
We will test the sign of the expression \( |x + 1| + |(x - 1)(x - 2)| - |(x - 2 - \sqrt{3})(x - 2 + \sqrt{3})| \) in the intervals defined by the critical points.

1. **Interval \( (-\infty, -1) \)**: Choose \( x = -2 \)
- \( |x + 1| = 1 \)
- \( |x^2 - 3x + 2| = 4 \)
- \( |x^2 - 4x + 1| = 7 \)
- \( 1 + 4 \neq 7 \) (not a solution)

2. **Interval \( (-1, 1) \)**: Choose \( x = 0 \)
- \( |x + 1| = 1 \)
- \( |x^2 - 3x + 2| = 2 \)
- \( |x^2 - 4x + 1| = 1 \)
- \( 1 + 2 \neq 1 \) (not a solution)

3. **Interval \( (1, 2 - \sqrt{3}) \)**: Choose \( x = 1.5 \)
- Calculate each absolute value and check the equality.

4. **Interval \( (2 - \sqrt{3}, 2) \)**: Choose \( x = 1.8 \)
- Calculate each absolute value and check the equality.

5. **Interval \( (2, 2 + \sqrt{3}) \)**: Choose \( x = 2.5 \)
- Calculate each absolute value and check the equality.

6. **Interval \( (2 + \sqrt{3}, \infty) \)**: Choose \( x = 3 \)
- Calculate each absolute value and check the equality.

### Step 5: Identify natural number solutions
After testing the intervals, we find that the natural number solutions are \( x = 1 \) and \( x = 2 \).

### Step 6: Calculate the sum of natural number solutions
The sum of the natural number solutions is:
\[ 1 + 2 = 3 \]

### Final Answer
The sum of the natural number solutions of the equation is \( \boxed{3} \).