The numberof real values of x that satisfies the equation `x^(4)+4x^(3)+12x^(2)+7x-3=0` is

Answer

To determine the number of real values of \( x \) that satisfy the equation

\[ x^4 + 4x^3 + 12x^2 + 7x - 3 = 0, \]

we will analyze the function step by step.

### Step 1: Define the function
Let

\[ f(x) = x^4 + 4x^3 + 12x^2 + 7x - 3. \]

### Step 2: Find the first derivative
We differentiate \( f(x) \) to find the critical points:

\[ f'(x) = 4x^3 + 12x^2 + 24x + 7. \]

### Step 3: Find the second derivative
Next, we differentiate \( f'(x) \) to analyze the concavity:

\[ f''(x) = 12x^2 + 24x + 24. \]

### Step 4: Analyze the second derivative
We can factor out the common term from \( f''(x) \):

\[ f''(x) = 12(x^2 + 2x + 2). \]

Now, we need to find the discriminant of the quadratic \( x^2 + 2x + 2 \):

\[ D = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 2 = 4 - 8 = -4. \]

Since the discriminant \( D < 0 \), the quadratic has no real roots, meaning \( f''(x) > 0 \) for all \( x \). Thus, \( f'(x) \) is always increasing.

### Step 5: Analyze the first derivative
Since \( f'(x) \) is a cubic polynomial, it can have at most 3 real roots. However, since \( f'(x) \) is always increasing, it can have at most one real root.

### Step 6: Evaluate the function at specific points
We will evaluate \( f(x) \) at some specific points to determine the behavior of the function:

1. **At \( x = 0 \)**:
\[ f(0) = 0^4 + 4(0)^3 + 12(0)^2 + 7(0) - 3 = -3. \]

2. **At \( x = 1 \)**:
\[ f(1) = 1^4 + 4(1)^3 + 12(1)^2 + 7(1) - 3 = 1 + 4 + 12 + 7 - 3 = 21. \]

3. **At \( x = -5 \)**:
\[ f(-5) = (-5)^4 + 4(-5)^3 + 12(-5)^2 + 7(-5) - 3 = 625 - 500 + 300 - 35 - 3 = 387. \]

### Step 7: Determine the number of real roots
From the evaluations:
- \( f(0) = -3 \) (negative)
- \( f(1) = 21 \) (positive)
- \( f(-5) = 387 \) (positive)

Since \( f(0) < 0 \) and \( f(1) > 0 \), by the Intermediate Value Theorem, there is at least one root in the interval \( (0, 1) \).

Since \( f'(x) \) has at most one real root, \( f(x) \) can change from negative to positive only once. Therefore, there can be at most 2 real roots.

### Conclusion
Thus, the number of real values of \( x \) that satisfy the equation \( x^4 + 4x^3 + 12x^2 + 7x - 3 = 0 \) is:

\[ \boxed{2}. \]