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The image shows a mathematical exercise written in Indonesian, asking to use the "Horner's Method" to find the quotient and remainder from the division of \( P(x) \) by \( Q(x) \) for various polynomials. The specific questions circled in the image are (c), (e), (i), and (j), with different forms of polynomials given.

Let's start by addressing question (c):

### Problem:
- \( P(x) = x^5 - 2x^3 + 7x - 1 \)
- \( Q(x) = x + 2 \)

Using **Horner's Method**, we want to divide \( P(x) \) by \( Q(x) \) and find both the quotient and remainder.

### Steps:
1. **Set the divisor** \( Q(x) = x + 2 \), so the root is \( -2 \).
2. **Write the coefficients of \( P(x) \)**: \( [1, 0, -2, 0, 7, -1] \).

Now, we can apply synthetic division to find the quotient and remainder.

Let me calculate that for you.For the division of \( P(x) = x^5 - 2x^3 + 7x - 1 \) by \( Q(x) = x + 2 \), using Horner's method, the result is:

- **Quotient**: \( x^4 - 2x^3 + 2x^2 - 4x + 15 \)
- **Remainder**: \( -31 \)

Would you like a detailed explanation of this process or help with another question from the exercise? 

Here are five related questions:
1. How does Horner's method simplify polynomial division?
2. What is the significance of the remainder in polynomial division?
3. Can synthetic division be used for any divisor other than \( x - r \)?
4. What are the applications of dividing polynomials in higher-level mathematics?
5. How does the degree of the quotient relate to the degrees of \( P(x) \) and \( Q(x) \)?

**Tip:** When performing synthetic division, always remember that missing terms (like \( x^4 \) in this case) should have a coefficient of zero.

Use Horner's Method to find the quotient and remainder of the division P(x) by Q(x).

Learn how to use Horner's Method to divide the polynomial P(x) by Q(x) and find both the quotient and remainder. This problem includes polynomials of various degrees and is suitable for students in Grades 10-12.

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