1. **State the problem:** Find a polynomial function $f(x)$ of degree 4 with real zeros $-6, 0, 3, 2$ and that passes through the point $\left(-\frac{1}{2}, -385\right)$. 2. **Write the general form:** Since the zeros are $-6, 0, 3, 2$, the polynomial can be written as $$f(x) = a(x + 6)(x)(x - 3)(x - 2)$$ where $a$ is a constant to be determined. 3. **Use the given point to find $a$:** Substitute $x = -\frac{1}{2}$ and $f(x) = -385$ into the equation: $$-385 = a\left(-\frac{1}{2} + 6\right)\left(-\frac{1}{2}\right)\left(-\frac{1}{2} - 3\right)\left(-\frac{1}{2} - 2\right)$$ 4. **Simplify inside the parentheses:** $$-385 = a\left(\frac{11}{2}\right)\left(-\frac{1}{2}\right)\left(-\frac{7}{2}\right)\left(-\frac{5}{2}\right)$$ 5. **Multiply the factors:** $$\left(\frac{11}{2}\right) \times \left(-\frac{1}{2}\right) = -\frac{11}{4}$$ $$-\frac{11}{4} \times \left(-\frac{7}{2}\right) = \frac{77}{8}$$ $$\frac{77}{8} \times \left(-\frac{5}{2}\right) = -\frac{385}{16}$$ 6. **Substitute back:** $$-385 = a \times \left(-\frac{385}{16}\right)$$ 7. **Solve for $a$:** $$-385 = -\frac{385}{16} a$$ Divide both sides by $-\frac{385}{16}$: $$a = \frac{-385}{-\frac{385}{16}} = \cancel{-385} \times \frac{16}{\cancel{385}} = 16$$ 8. **Write the final polynomial:** $$f(x) = 16(x + 6)(x)(x - 3)(x - 2)$$ **Answer:** $$\boxed{f(x) = 16x(x + 6)(x - 3)(x - 2)}$$