Assertion(A): `x^(2)+x+1 gt 0` for all positive real values of x only.
Reason (R) : When `b^(2)-4ac lt 0, a , ax^(2)+bx+x` have same sign for all real values of x.

Options

A. * Both A, R are true and R explain Assertion
B. * Both A, R are true but R does't explain A
C. * A is true R is false
D. * A is false R is true

Answer

To solve the given question, we will analyze both the assertion and the reason step by step.

### Step 1: Analyze the Assertion
The assertion states that \( x^2 + x + 1 > 0 \) for all positive real values of \( x \).

1. **Evaluate the quadratic expression**:
- The quadratic expression is \( x^2 + x + 1 \).
- We can check if this expression is always positive for positive values of \( x \) by analyzing its discriminant.

2. **Calculate the discriminant**:
- The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c \) is given by \( D = b^2 - 4ac \).
- Here, \( a = 1 \), \( b = 1 \), and \( c = 1 \).
- Thus, \( D = 1^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \).

3. **Interpret the discriminant**:
- Since the discriminant is negative (\( D < 0 \)), the quadratic does not have real roots and opens upwards (as \( a > 0 \)).
- Therefore, \( x^2 + x + 1 > 0 \) for all real values of \( x \), including all positive values of \( x \).

### Conclusion for Assertion
The assertion is true: \( x^2 + x + 1 > 0 \) for all positive real values of \( x \).

### Step 2: Analyze the Reason
The reason states that when \( b^2 - 4ac < 0 \), the quadratic \( ax^2 + bx + c \) has the same sign for all real values of \( x \).

1. **Understanding the statement**:
- The condition \( b^2 - 4ac < 0 \) implies that the quadratic does not intersect the x-axis, meaning it does not have real roots.
- If \( a > 0 \), the quadratic opens upwards and is positive for all \( x \).
- If \( a < 0 \), the quadratic opens downwards and is negative for all \( x \).

2. **Apply to our case**:
- In our case, since \( a = 1 > 0 \) and \( D < 0 \), it confirms that \( x^2 + x + 1 \) is positive for all real \( x \).

### Conclusion for Reason
The reason is true: when \( b^2 - 4ac < 0 \), the quadratic has the same sign for all real values of \( x \).

### Final Conclusion
Both the assertion and the reason are true, and the reason correctly explains the assertion. Therefore, the correct option is: **Both A and R are true, and R explains A.**