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To solve this problem, let’s analyze the given dataTo solve this problem, let’s analyze the given 

1. **Degree**: 4, meaning the polynomial \( f(x) \) is of degree 4.
2. **Zeros**:
   - \( -5 + 2i \) is a zero.
   - Since the polynomial has real coefficients, the complex conjugate \( -5 - 2i \) must also be a zero.
   - \( -1 \) with multiplicity 2 is another zero.

### Step 1: Write the factors corresponding to each zero
For each zero \( c \), a corresponding factor is \( (x - c) \).

1. For \( -5 + 2i \): the factor is \( (x + 5 - 2i) \).
2. For \( -5 - 2i \): the factor is \( (x + 5 + 2i) \).
3. For \( -1 \) with multiplicity 2: the factor is \( (x + 1)^2 \).

### Step 2: Form the polynomial \( f(x) \)
The polynomial can be represented as:
\[
f(x) = a \cdot (x + 5 - 2i)(x + 5 + 2i)(x + 1)^2
\]
where \( a \) is the leading coefficient.

### Step 3: Simplify the complex factors
Use the difference of squares to simplify \( (x + 5 - 2i)(x + 5 + 2i) \):
\[
(x + 5 - 2i)(x + 5 + 2i) = (x + 5)^2 - (2i)^2
\]
Since \( (2i)^2 = -4 \), this simplifies to:
\[
(x + 5)^2 + 4
\]

Expanding \( (x + 5)^2 + 4 \):
\[
(x + 5)^2 + 4 = x^2 + 10x + 25 + 4 = x^2 + 10x + 29
\]

### Step 4: Substitute and expand the polynomial
Now we have:
\[
f(x) = a \cdot (x^2 + 10x + 29)(x + 1)^2
\]

Expanding \( (x + 1)^2 \):
\[
(x + 1)^2 = x^2 + 2x + 1
\]

Finally, multiply these expressions:
\[
f(x) = a \cdot (x^2 + 10x + 29)(x^2 + 2x + 1)
\]

Expanding this product:
\[
f(x) = a \cdot (x^4 + 2x^3 + x^2 + 10x^3 + 20x^2 + 10x + 29x^2 + 58x + 29)
\]

Combine like terms:
\[
f(x) = a \cdot (x^4 + 12x^3 + 50x^2 + 68x + 29)
\]

Thus, the polynomial \( f(x) \) is:
\[
f(x) = a(x^4 + 12x^3 + 50x^2 + 68x + 29)
\]

This is the polynomial in terms of the variable \( a \) as the leading coefficient.

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

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### Related Questions
1. What is the difference between real and complex roots in polynomials?
2. How do you determine the degree of a polynomial given its roots?
3. How can you find the multiplicity of a root?
4. What happens to a polynomial if a root has a higher multiplicity?
5. How does the leading coefficient \( a \) affect the shape of the polynomial graph?

### Tip
When a polynomial has real coefficients, any complex roots must occur in conjugate pairs.

Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree 4; zeros: -5 + 2i; -1 multiplicity 2.

Learn to form a degree 4 polynomial with real coefficients using given zeros, including complex roots with conjugates and roots with multiplicity. This problem demonstrates the conjugate root theorem and uses polynomial expansion techniques.

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