To analyze the given cubic function \( f(x) = ax^3 + bx^2 + cx + d \) and its behavior based on the provided derivative table, we can extract critical information:

1. **Critical Points:**
- The first derivative \( f'(x) \) changes from positive to zero at \( x = -2 \), indicating a local maximum, since \( f'(x) \) changes from positive to negative.
- It also changes from negative to zero at \( x = 0 \), indicating a local minimum.

2. **Behavior at Infinity:**
- As \( x \to -\infty \), \( f(x) \to -\infty \) suggests \( a < 0 \).
- As \( x \to +\infty \), \( f(x) \to +\infty \) indicates \( a > 0 \).

Given these conditions, we can derive:
- \( a \) cannot simultaneously be both greater than and less than zero, so we have \( a < 0 \).

3. **Values of \( f(x) \) at Critical Points:**
- At the local maximum \( f(-2) = 2 \) and at the local minimum \( f(0) = 1 \).

4. **Coefficients Summary:**
- We can conclude there will be inconsistencies if \( a \) is not negative.

Thus, to answer your question about the number of possible values for \( a, b, c, d \), given the constraints extracted from the behavior of \( f(x) \) and its derivative \( f'(x) \), it seems there may be **limited solutions**. Therefore, the range of coefficients can be placed under examination, revealing **2 distinct ranges** likely due to sign restrictions imposed by polynomial behavior (as suggested by the local max and min).

Overall, in your question, the analysis indicates **two unique numbers can represent coefficients \( a, b, c, d \)**. Thus, the answer is **2**.