Statement-1: `If x^(2) + ax + 4 gt 0 "for all" x in R`, then `a in (-4, 4)`.
Statement-2: The sign of quadratic expression `ax^(2) + bx + c` is always same as that of 'a' except for those values of x which lie between its roots.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Explanation

To solve the problem, we need to analyze both statements provided and determine their validity step by step.

### Step 1: Analyze Statement 1
**Statement 1:** If \( x^2 + ax + 4 > 0 \) for all \( x \in \mathbb{R} \), then \( a \in (-4, 4) \).

1. **Condition for a Quadratic to be Positive:**
For the quadratic expression \( x^2 + ax + 4 \) to be greater than zero for all real numbers \( x \), two conditions must be satisfied:
- The coefficient of \( x^2 \) (which is 1) must be greater than zero.
- The discriminant of the quadratic must be less than zero.

2. **Calculate the Discriminant:**
The discriminant \( D \) of the quadratic \( x^2 + ax + 4 \) is given by:
\[ D = b^2 - 4ac = a^2 - 4 \cdot 1 \cdot 4 = a^2 - 16 \]

3. **Set the Discriminant Condition:**
For the quadratic to be positive for all \( x \), we need:
\[ D < 0 \implies a^2 - 16 < 0 \implies a^2 < 16 \]

4. **Solve the Inequality:**
Taking the square root of both sides gives:
\[ -4 < a < 4 \]
Thus, \( a \in (-4, 4) \).

### Conclusion for Statement 1:
Statement 1 is **True**.

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### Step 2: Analyze Statement 2
**Statement 2:** The sign of the quadratic expression \( ax^2 + bx + c \) is always the same as that of \( a \) except for those values of \( x \) which lie between its roots.

1. **Understanding Quadratic Behavior:**
- If \( a > 0 \), the parabola opens upwards. The quadratic will be positive outside the roots and negative between the roots.
- If \( a < 0 \), the parabola opens downwards. The quadratic will be negative outside the roots and positive between the roots.

2. **Conclusion for Statement 2:**
The statement correctly describes the behavior of quadratic functions. Thus, Statement 2 is also **True**.

### Final Conclusion:
- **Statement 1 is True.**
- **Statement 2 is True.**
- However, Statement 2 does not serve as a correct explanation for Statement 1.

### Answer:
The correct option is: **Second option: Statement 1 is true, Statement 2 is true, and Statement 2 is not a correct explanation for Statement 1.**

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