Determine the sign of the expression
i) `x^(2)+x+1` ii) `-x^(2)+x-1` for `x in R`.

Answer

To determine the sign of the expressions \( i) \, x^2 + x + 1 \) and \( ii) \, -x^2 + x - 1 \) for \( x \in \mathbb{R} \), we will analyze each expression using the properties of quadratic functions.

### Step-by-Step Solution:

#### Part i: \( x^2 + x + 1 \)

1. **Identify coefficients**:
- The expression is in the form \( ax^2 + bx + c \).
- Here, \( a = 1 \), \( b = 1 \), and \( c = 1 \).

2. **Calculate the discriminant \( D \)**:
- The discriminant \( D \) is given by \( D = b^2 - 4ac \).
- Substituting the values, we get:
\[ D = 1^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]

3. **Analyze the sign of \( a \) and \( D \)**:
- Since \( a = 1 > 0 \) and \( D = -3 < 0 \), we can use the properties of quadratic functions.
- For a quadratic function where \( a > 0 \) and \( D < 0 \), the function is positive for all \( x \in \mathbb{R} \).

4. **Conclusion for part i**:
- Therefore, \( x^2 + x + 1 > 0 \) for all \( x \in \mathbb{R} \).

#### Part ii: \( -x^2 + x - 1 \)

1. **Identify coefficients**:
- The expression is also in the form \( ax^2 + bx + c \).
- Here, \( a = -1 \), \( b = 1 \), and \( c = -1 \).

2. **Calculate the discriminant \( D \)**:
- Again, we use \( D = b^2 - 4ac \).
- Substituting the values, we get:
\[ D = 1^2 - 4 \cdot (-1) \cdot (-1) = 1 - 4 = -3 \]

3. **Analyze the sign of \( a \) and \( D \)**:
- Here, \( a = -1 < 0 \) and \( D = -3 < 0 \).
- For a quadratic function where \( a < 0 \) and \( D < 0 \), the function is negative for all \( x \in \mathbb{R} \).

4. **Conclusion for part ii**:
- Therefore, \( -x^2 + x - 1 < 0 \) for all \( x \in \mathbb{R} \).

### Final Answers:
- i) \( x^2 + x + 1 > 0 \) for all \( x \in \mathbb{R} \)
- ii) \( -x^2 + x - 1 < 0 \) for all \( x \in \mathbb{R} \)