If `b^2>4ac`, then `a(x^2 +4x+4)^2+b(x^2 +4x +4)+c=0` has distinct real roots if :

Options

A. a and b are of same sign
B. b and c are of same sign
C. a and c are of same sign
D. a,b and c are of same sign

Answer

To solve the problem, we need to analyze the quadratic equation given in the form:

\[ a(x^2 + 4x + 4)^2 + b(x^2 + 4x + 4) + c = 0 \]

First, we can simplify the expression \( x^2 + 4x + 4 \):

1. **Step 1: Recognize the perfect square.**
\[ x^2 + 4x + 4 = (x + 2)^2 \]
Thus, we can rewrite the equation as:
\[ a((x + 2)^2)^2 + b((x + 2)^2) + c = 0 \]

2. **Step 2: Substitute \( t = (x + 2)^2 \).**
The equation now becomes:
\[ at^2 + bt + c = 0 \]
where \( t \geq 0 \) (since \( t \) is a square term).

3. **Step 3: Determine the conditions for distinct real roots.**
For the quadratic equation \( at^2 + bt + c = 0 \) to have distinct real roots, the discriminant must be greater than zero:
\[ D = b^2 - 4ac > 0 \]

4. **Step 4: Analyze the conditions given.**
We are given that \( b^2 > 4ac \). This implies that the quadratic in \( t \) has distinct real roots.

5. **Step 5: Consider the behavior of the quadratic function.**
- If \( a > 0 \), the parabola opens upwards. For it to have distinct real roots, it must intersect the t-axis at two points, which means the vertex must be below the t-axis.
- If \( a < 0 \), the parabola opens downwards. For it to have distinct real roots, the vertex must be above the t-axis.

6. **Step 6: Analyze the signs of \( a \), \( b \), and \( c \).**
- If \( a > 0 \), the vertex (given by \( -\frac{b}{2a} \)) must be greater than zero for the roots to be valid in the domain \( t \geq 0 \):
\[ -\frac{b}{2a} \geq 0 \implies b \leq 0 \]
- If \( a < 0 \), the vertex must be less than zero:
\[ -\frac{b}{2a} < 0 \implies b > 0 \]

7. **Step 7: Determine the signs of \( c \).**
- If \( a > 0 \) and \( b < 0 \), then \( c \) must also be positive for the quadratic to cross the t-axis twice.
- If \( a < 0 \) and \( b > 0 \), then \( c \) must be negative.

8. **Step 8: Conclude the conditions.**
- Therefore, the conditions for distinct real roots can be summarized as:
- \( a \) and \( c \) must have the same sign.
- \( b \) must be of opposite sign to \( a \).

Thus, the correct answer is that \( a \) and \( c \) must have the same sign.