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To solve this problem using Descartes' Rule of Signs, we need to analyze the function \( f(x) = x^3 + 5x^2 + 6x + 3 \) for changes in the signs of its terms.

### Step 1: Determine Positive Real Zeros
To find the possible number of positive real zeros, we look at \( f(x) \) and count the sign changes between terms:

1. \( x^3 \) (positive),
2. \( +5x^2 \) (positive),
3. \( +6x \) (positive),
4. \( +3 \) (positive).

There are **no sign changes** in \( f(x) \). According to Descartes' Rule of Signs, this implies there are **0 possible positive real zeros**.

### Conclusion
The possible number of positive real zeros for the function \( f(x) = x^3 + 5x^2 + 6x + 3 \) is:

**Answer:** 0

Use Descartes' Rule of Signs to determine the possible numbers of positive and negative real zeros of f(x) = x^3 + 5x^2 + 6x + 3. What are the possible numbers of positive real zeros?

Determine the possible number of positive real zeros for the function f(x) = x^3 + 5x^2 + 6x + 3 using Descartes' Rule of Signs. This problem covers polynomial functions and sign-change analysis. Suitable for high school students in Grades 10-12.

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