The number of points, at which the function `f(x)=|2x+1|-3|x+2|+|x^(2)+x-2|, x in R` is not differentiable, is _______.

Answer

To determine the number of points at which the function \( f(x) = |2x + 1| - 3|x + 2| + |x^2 + x - 2| \) is not differentiable, we need to analyze the points where the absolute value expressions change.

### Step-by-step Solution:

1. **Identify the critical points from each absolute value term:**
- For \( |2x + 1| \):
- Set \( 2x + 1 = 0 \) → \( x = -\frac{1}{2} \)
- For \( |x + 2| \):
- Set \( x + 2 = 0 \) → \( x = -2 \)
- For \( |x^2 + x - 2| \):
- Factor \( x^2 + x - 2 = (x + 2)(x - 1) \)
- Set \( x^2 + x - 2 = 0 \) → \( x = -2 \) and \( x = 1 \)

2. **List all critical points:**
- From \( |2x + 1| \): \( x = -\frac{1}{2} \)
- From \( |x + 2| \): \( x = -2 \)
- From \( |x^2 + x - 2| \): \( x = -2 \) and \( x = 1 \)

Thus, the critical points are \( -2, -\frac{1}{2}, 1 \).

3. **Check for differentiability at each critical point:**
- A function is not differentiable at points where the absolute value expressions change, which occur at the critical points identified.

4. **Evaluate the differentiability at \( x = -2 \):**
- Since \( |x + 2| \) appears twice in the expression, we need to check if the function is differentiable at this point.
- Calculate the left-hand and right-hand derivatives at \( x = -2 \):
- For \( x < -2 \):
\[ f(x) = -(2x + 1) - 3(-x - 2) + (x^2 + x - 2) = -2x - 1 + 3x + 6 + x^2 + x - 2 = x^2 + 2x + 3 \]
The derivative is \( f'(x) = 2x + 2 \).
- For \( x > -2 \):
\[ f(x) = -(2x + 1) - 3(x + 2) + (x^2 + x - 2) = -2x - 1 - 3x - 6 + x^2 + x - 2 = x^2 - 4x - 9 \]
The derivative is \( f'(x) = 2x - 4 \).
- Evaluate at \( x = -2 \):
- Left-hand derivative: \( f'(-2) = 2(-2) + 2 = -4 + 2 = -2 \)
- Right-hand derivative: \( f'(-2) = 2(-2) - 4 = -4 - 4 = -8 \)
- Since the left-hand and right-hand derivatives are not equal, \( f(x) \) is not differentiable at \( x = -2 \).

5. **Conclusion:**
- The points where \( f(x) \) is not differentiable are \( x = -\frac{1}{2}, 1, \) and \( -2 \).
- However, since \( -2 \) is a repeated critical point, we only count it once.

### Final Answer:
The number of points at which the function \( f(x) \) is not differentiable is **3**.