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You are tasked with finding a polynomial \( f(x) \) of degree 3 with real coefficients, given the following zeros: \( -3 \) and \( 2 + i \).

### Step-by-step Solution:

1. **Understanding the zeros:**
   - One zero is \( -3 \).
   - The second zero is \( 2 + i \).
   - Since the polynomial must have real coefficients, the complex conjugate of \( 2 + i \), which is \( 2 - i \), must also be a zero.

2. **Constructing the polynomial:**
   The polynomial can be written as:
   \[
   f(x) = (x - (-3))(x - (2 + i))(x - (2 - i))
   \]
   This simplifies to:
   \[
   f(x) = (x + 3)(x - (2 + i))(x - (2 - i))
   \]

3. **Simplifying the product of the complex conjugates:**
   We can simplify \( (x - (2 + i))(x - (2 - i)) \) using the difference of squares formula:
   \[
   (x - (2 + i))(x - (2 - i)) = (x - 2)^2 - (i)^2 = (x - 2)^2 + 1
   \]
   Expanding this:
   \[
   (x - 2)^2 + 1 = (x^2 - 4x + 4) + 1 = x^2 - 4x + 5
   \]

4. **Expanding the full polynomial:**
   Now, multiply this result by \( (x + 3) \):
   \[
   f(x) = (x + 3)(x^2 - 4x + 5)
   \]
   Expanding this:
   \[
   f(x) = x(x^2 - 4x + 5) + 3(x^2 - 4x + 5)
   \]
   \[
   f(x) = x^3 - 4x^2 + 5x + 3x^2 - 12x + 15
   \]
   Simplifying:
   \[
   f(x) = x^3 - x^2 - 7x + 15
   \]

Thus, the polynomial is:
\[
f(x) = x^3 - x^2 - 7x + 15
\]

Would you like further clarification or details on any part of this process? 

### Related questions:
1. What are the complex conjugates and why are they important for polynomials with real coefficients?
2. How does the difference of squares formula work for complex numbers?
3. Can the polynomial be factorized further, or is it already in its simplest form?
4. How would the graph of this cubic polynomial look based on its zeros?
5. What is the behavior of a cubic polynomial at infinity?

### Tip:
When dealing with polynomials that have complex zeros, always remember that the conjugate of any complex zero must also be included if the coefficients are real.

Find a polynomial f(x) of degree 3 with real coefficients and the following zeros: -3, 2 + i.

Learn how to find a degree 3 polynomial with real coefficients, given the zeros -3, 2 + i. This problem involves polynomial expansion, the complex conjugate root theorem, and difference of squares. Suitable for Grades 10-12.

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