For `x^2-(a+3)|x|-4=0` to have real solutions, the range of `a` is
a)`(-oo,-7]uu[1,oo)`
b)`(-3,oo)`
c)`(-oo,-7]`
d)`[1,oo)`

Options

A. `(-oo,-7]cup [1,oo)`
B. `(-3,oo)`
C. `(-oo,-7]`
D. `[1,oo)`

Explanation

To solve the equation \( x^2 - (a+3)|x| - 4 = 0 \) for real solutions, we need to analyze the expression and find the range of \( a \).

### Step 1: Rewrite the equation
We start with the equation:
\[ x^2 - (a+3)|x| - 4 = 0 \]
This can be rearranged as:
\[ x^2 - 4 = (a+3)|x| \]

### Step 2: Analyze the left-hand side
The left-hand side, \( x^2 - 4 \), can be factored:
\[ (x - 2)(x + 2) = (a + 3)|x| \]
The expression \( x^2 - 4 \) is a quadratic function that opens upwards and has roots at \( x = -2 \) and \( x = 2 \). It is negative between these roots and positive outside this interval.

### Step 3: Determine conditions for real solutions
For the equation to have real solutions, the right-hand side, \( (a + 3)|x| \), must be non-negative. This means:
\[ x^2 - 4 \geq 0 \quad \text{or} \quad |x| \geq \frac{x^2 - 4}{a + 3} \]
The left-hand side is non-negative when \( x \leq -2 \) or \( x \geq 2 \).

### Step 4: Analyze the critical points
At the critical points \( x = -2 \) and \( x = 2 \):
1. For \( x = 2 \):
\[ 2^2 - 4 = 0 \implies (a + 3) \cdot 2 = 0 \implies a + 3 = 0 \implies a = -3 \]
2. For \( x = -2 \):
\[ (-2)^2 - 4 = 0 \implies (a + 3) \cdot 2 = 0 \implies a + 3 = 0 \implies a = -3 \]

### Step 5: Determine the minimum value of \( a \)
To ensure that the equation has real solutions, we need to ensure that \( a + 3 \) is non-negative when \( |x| \) is at its minimum, which occurs at the critical points. Thus, we find:
\[ a + 3 \geq 0 \implies a \geq -3 \]

### Step 6: Analyze the behavior of the function
As \( |x| \) increases beyond 2, \( (a + 3)|x| \) must also be able to match the increasing values of \( x^2 - 4 \). Therefore, we also consider:
\[ a + 3 > 0 \implies a > -3 \]

### Conclusion
Thus, for the equation to have real solutions, the range of \( a \) is:
\[ a \geq -3 \]
However, since we need \( a + 3 \) to be positive for all \( x \), we find that the minimum value of \( a \) is \( -3 \).

### Final Answer
The correct option is:
**b) (-3, ∞)**