The problem asks for the polynomial function of the lowest degree with real coefficients that has the zeros $0, -3, 1 + 3i$.

Step-by-step solution:

1. Given Zeros: The roots provided are $0$, $-3$, and $1 + 3i$. Since the polynomial must have real coefficients and $1 + 3i$ is a complex number, its conjugate, $1 - 3i$, must also be a root.

2. Forming the Polynomial: The polynomial's factors will be: $f(x) = x(x + 3)\left(x - (1 + 3i)\right)\left(x - (1 - 3i)\right)$

3. Simplifying the Complex Factors: We can simplify the product of the two complex factors, using the difference of squares: $\left(x - (1 + 3i)\right)\left(x - (1 - 3i)\right) = \left[(x - 1)^2 - (3i)^2\right] = (x - 1)^2 + 9 = x^2 - 2x + 10$

4. Expanding the Full Polynomial: Now, we have the polynomial: $f(x) = x(x + 3)(x^2 - 2x + 10)$ First, expand $x(x + 3)$: $x(x + 3) = x^2 + 3x$ Now expand the entire expression: $f(x) = (x^2 + 3x)(x^2 - 2x + 10)$ Multiply term by term: $f(x) = x^2(x^2 - 2x + 10) + 3x(x^2 - 2x + 10)$ $f(x) = x^4 - 2x^3 + 10x^2 + 3x^3 - 6x^2 + 30x$ Combine like terms: $f(x) = x^4 + (3x^3 - 2x^3) + (10x^2 - 6x^2) + 30x$ $f(x) = x^4 + x^3 + 4x^2 + 30x$

5. Final Answer: The polynomial is $f(x) = x^4 + x^3 + 4x^2 + 30x$, which matches option B.

The correct answer is B.

Would you like further clarification or additional details?

Here are 5 related questions:

1. How do you determine the degree of a polynomial from its roots?
2. Why do complex roots appear in conjugate pairs for polynomials with real coefficients?
3. What is the general form of a quadratic factor when given a complex root?
4. How do you expand polynomials using the distributive property?
5. Can a polynomial have more than one degree-4 form with the same set of zeros?

Tip: When working with complex roots, always remember that their conjugates must also be included to ensure the polynomial has real coefficients.