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You have uploaded a math problem asking to find the quotient and remainder when dividing:

\[ 2x^4 + 3x^3 - 12x^2 - 13x + 5 \]

by:

\[ 2x^2 + 3x - 2 \]

using Horner's method.

### Step-by-step solution using Horner's method for polynomial division:

1. **First, set up the division.**
   - Dividend: \( 2x^4 + 3x^3 - 12x^2 - 13x + 5 \)
   - Divisor: \( 2x^2 + 3x - 2 \)

   This is a synthetic division process adapted for quadratic division.

2. **We start by focusing on the degree of the divisor** (the highest degree of \(x^2\)), and we align the coefficients for the synthetic division.

Now, let me perform the actual division and find the quotient and remainder.The quotient and remainder from dividing \( 2x^4 + 3x^3 - 12x^2 - 13x + 5 \) by \( 2x^2 + 3x - 2 \) using synthetic division (Horner's method) are:

- **Quotient**: \( x^2 - 5 \)
- **Remainder**: \( 2x - 5 \)

### Final expression:

\[
\frac{2x^4 + 3x^3 - 12x^2 - 13x + 5}{2x^2 + 3x - 2} = x^2 - 5 + \frac{2x - 5}{2x^2 + 3x - 2}
\]

Would you like a detailed explanation of the steps involved? Feel free to ask any questions!

Here are 5 related questions to further your understanding:

1. How do you apply synthetic division to polynomials with higher degree divisors?
2. What is the importance of the remainder in polynomial division?
3. Can Horner's method be applied to polynomials with complex roots?
4. How would you divide a polynomial by a cubic polynomial using synthetic division?
5. What happens when a polynomial division results in no remainder?

**Tip**: Horner's method can be adapted to work with higher-degree polynomials, but it requires careful setup of the divisor's coefficients.

Tentukan sisa dan hasil bagi 2x^4 + 3x^3 - 12x^2 - 13x + 5 dibagi (2x^2 + 3x - 2) Hitunglah menggunakan metode horner

Learn how to divide polynomials using Horner's method. In this example, we divide 2x^4 + 3x^3 - 12x^2 - 13x + 5 by 2x^2 + 3x - 2 to find the quotient and remainder. Suitable for students in Grades 10-12.

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