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Let's solve both questions in the image step-by-step.

### First Question:
**Write a polynomial function of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros: -4, 1, 2.**

The polynomial can be formed by setting each zero as a factor:
\[
f(x) = (x + 4)(x - 1)(x - 2)
\]

Let's expand this expression step-by-step.

1. Multiply \((x + 4)\) and \((x - 1)\):
   \[
   (x + 4)(x - 1) = x^2 + 3x - 4
   \]

2. Now, multiply \((x^2 + 3x - 4)\) by \((x - 2)\):
   \[
   f(x) = (x^2 + 3x - 4)(x - 2) = x^3 - 2x^2 + 3x^2 - 6x - 4x + 8 = x^3 + x^2 - 10x + 8
   \]

So, the correct polynomial is:
\[
f(x) = x^3 + x^2 - 10x + 8
\]

From the answer choices, this matches the last option:
\[
\boxed{f(x) = x^3 + x^2 - 10x + 8}
\]

### Second Question:
**Find all the zeros of the polynomial function: \( f(x) = x^4 - 5x^2 - 36 \).**

To find the zeros, observe that this polynomial can be solved by substitution:
1. Let \( u = x^2 \), so the equation becomes:
   \[
   u^2 - 5u - 36 = 0
   \]

2. Solve this quadratic equation using the quadratic formula:
   \[
   u = \frac{5 \pm \sqrt{(-5)^2 + 4 \cdot 36}}{2} = \frac{5 \pm \sqrt{25 + 144}}{2} = \frac{5 \pm \sqrt{169}}{2} = \frac{5 \pm 13}{2}
   \]

3. This yields:
   \[
   u = \frac{5 + 13}{2} = 9 \quad \text{and} \quad u = \frac{5 - 13}{2} = -4
   \]

4. Now, substitute back \( u = x^2 \):
   - For \( u = 9 \): \( x^2 = 9 \Rightarrow x = \pm 3 \)
   - For \( u = -4 \): \( x^2 = -4 \Rightarrow x = \pm 2i \) (imaginary solutions)

So, the zeros are:
- **Real zeros:** \( x = -3 \) and \( x = 3 \)
- **Imaginary zeros:** \( x = -2i \) and \( x = 2i \)

The final answer in numerical order:
- **Real:** \( x = -3, x = 3 \)
- **Imaginary:** \( x = -2i, x = 2i \)

---

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

### Additional Questions
1. How do you verify if a polynomial has a given set of roots?
2. How do you expand a polynomial with more than three factors?
3. Can complex roots occur without imaginary numbers in the polynomial?
4. What are other methods to find polynomial roots besides substitution?
5. How does the Rational Root Theorem help in finding rational roots of polynomials?

**Tip:** When expanding polynomials, simplify each step gradually to avoid mistakes with signs and coefficients.

Write a polynomial function of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros: -4, 1, 2. Also, find all the zeros of the polynomial function f(x) = x^4 - 5x^2 - 36.

Learn how to construct a polynomial function with specified zeros and rational coefficients, and how to find all real and imaginary zeros of f(x) = x^4 - 5x^2 - 36. This example demonstrates polynomial factorization and the application of the quadratic formula, ideal for students in Grades 10-12.

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