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### Solution for Question 5:
The question asks to find \( f(-2) \) for the polynomial \( f(x) = x^3 - 6x^2 + 11x - 6 \) using synthetic division.

**Steps:**
1. **Set up synthetic division:** The coefficients of the polynomial are \( 1, -6, 11, -6 \).
2. **Use \( -2 \) as the divisor.**
3. **Perform the synthetic division.**

#### Synthetic Division Steps:
- **First row (Coefficients):** \( 1, -6, 11, -6 \)
- **Divisor:** \( -2 \)

|  | 1 | -6 | 11 | -6 |
|---|---|---|---|---|
| -2 |   | -2 | 16 | -54 |
|    | 1 | -8 | 5 | -60 |

#### Explanation:
1. Bring down the first coefficient \( 1 \).
2. Multiply \( 1 \times (-2) = -2 \) and write it under \( -6 \).
3. Add \( -6 + (-2) = -8 \).
4. Multiply \( -8 \times (-2) = 16 \) and write it under \( 11 \).
5. Add \( 11 + 16 = 5 \).
6. Multiply \( 5 \times (-2) = -10 \) and write it under \( -6 \).
7. Add \( -6 + (-10) = -60 \).

The remainder is \( -60 \), so \( f(-2) = -60 \).

**Answer:** The correct option is \(-60\).

---

### Solution for Question 6:
The question asks to find \( f(k) \) for the polynomial \( f(x) = x^2 - x + 3 \) using the remainder theorem, where \( k = 3 - 2i \).

**Steps:**
1. **Evaluate \( f(3 - 2i) \) directly.**

#### Direct Substitution:
\[
f(3 - 2i) = (3 - 2i)^2 - (3 - 2i) + 3
\]

Calculating step-by-step:
1. \( (3 - 2i)^2 = 9 - 12i + 4i^2 = 9 - 12i - 4 = 5 - 12i \) (since \( i^2 = -1 \)).
2. \( -(3 - 2i) = -3 + 2i \).
3. Combine all terms: 
   \[
   f(3 - 2i) = (5 - 12i) + (-3 + 2i) + 3
   \]
4. Simplify:
   \[
   f(3 - 2i) = 5 - 3 + 3 - 12i + 2i = 5 - 10i
   \]

**Answer:** The correct option is \( 5 - 10i \).

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Would you like more details or have any questions?

### Related Questions:
1. How is synthetic division different from polynomial long division?
2. What is the Remainder Theorem and how is it applied?
3. How do you handle complex numbers in polynomial evaluation?
4. Can synthetic division be used for divisors with complex numbers?
5. How does the result of synthetic division help in finding roots?

### Tip:
For polynomials, synthetic division is most useful when the divisor is a linear factor of the form \( x - c \).

Use synthetic division and remainder theorem to find f(-2) for f(x) = x^3 - 6x^2 + 11x - 6. Also, find f(k) for f(x) = x^2 - x + 3 with k = 3 - 2i.

Learn how to solve polynomial functions using synthetic division and the remainder theorem. The problem involves finding f(-2) for a cubic polynomial and evaluating a quadratic polynomial for a complex number. Ideal for high school students.

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