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The problem asks for the result of dividing the polynomial \(3x^4 + 2x^3 - 4x - 1\) by \(x + 2\). We can solve this using polynomial long division.

Let's go through the division process step-by-step:

1. **Divide the leading terms:**
   \[
   \frac{3x^4}{x} = 3x^3
   \]
   Multiply \(3x^3\) by \(x + 2\):
   \[
   3x^3 \cdot (x + 2) = 3x^4 + 6x^3
   \]
   Subtract this from the original polynomial:
   \[
   (3x^4 + 2x^3 - 4x - 1) - (3x^4 + 6x^3) = -4x^3 - 4x - 1
   \]

2. **Divide the new leading terms:**
   \[
   \frac{-4x^3}{x} = -4x^2
   \]
   Multiply \(-4x^2\) by \(x + 2\):
   \[
   -4x^2 \cdot (x + 2) = -4x^3 - 8x^2
   \]
   Subtract this from the current polynomial:
   \[
   (-4x^3 - 4x - 1) - (-4x^3 - 8x^2) = 8x^2 - 4x - 1
   \]

3. **Divide the new leading terms:**
   \[
   \frac{8x^2}{x} = 8x
   \]
   Multiply \(8x\) by \(x + 2\):
   \[
   8x \cdot (x + 2) = 8x^2 + 16x
   \]
   Subtract this from the current polynomial:
   \[
   (8x^2 - 4x - 1) - (8x^2 + 16x) = -20x - 1
   \]

4. **Divide the new leading terms:**
   \[
   \frac{-20x}{x} = -20
   \]
   Multiply \(-20\) by \(x + 2\):
   \[
   -20 \cdot (x + 2) = -20x - 40
   \]
   Subtract this from the current polynomial:
   \[
   (-20x - 1) - (-20x - 40) = 39
   \]

Thus, the quotient is \(3x^3 - 4x^2 + 8x - 20\), and the remainder is \(39\). The correct answer is **a. \(3x^3 - 4x^2 + 8x - 20\)**.

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Do you need further details or have any questions? Here are 5 related questions to deepen your understanding:

1. How can synthetic division be used to divide polynomials when the divisor is linear?
2. What happens to the remainder when dividing by a polynomial that has no real roots?
3. How can you check the result of polynomial division for accuracy?
4. What are the applications of polynomial division in real-world problems?
5. How does polynomial division relate to the Factor Theorem?

**Tip:** Always verify your division result by multiplying the quotient by the divisor and adding the remainder to see if you get back the original polynomial.

Hasil bagi suku banyak 3x^4 + 2x^3 - 4x - 1 oleh (x + 2) adalah?

Explore how to divide the polynomial 3x^4 + 2x^3 - 4x - 1 by the binomial (x + 2) using polynomial long division. Learn key algebraic concepts and formulas that are essential for polynomial division. Suitable for high school students in Grades 10-12.

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