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To solve the polynomial division problem, we need to divide \( 15x^2 + 14x - 8 \) by \( 3x + 4 \) using long division.

### Steps for Polynomial Long Division:

1. **Divide the first term** of the dividend by the first term of the divisor:
   \[
   \frac{15x^2}{3x} = 5x
   \]
   So, the first term of the quotient is \( 5x \).

2. **Multiply** \( 5x \) by \( 3x + 4 \):
   \[
   5x(3x + 4) = 15x^2 + 20x
   \]

3. **Subtract** this result from the original polynomial:
   \[
   (15x^2 + 14x - 8) - (15x^2 + 20x) = -6x - 8
   \]

4. **Divide** the first term of the new polynomial by the first term of the divisor:
   \[
   \frac{-6x}{3x} = -2
   \]
   So, the next term in the quotient is \( -2 \).

5. **Multiply** \( -2 \) by \( 3x + 4 \):
   \[
   -2(3x + 4) = -6x - 8
   \]

6. **Subtract** this from the remainder:
   \[
   (-6x - 8) - (-6x - 8) = 0
   \]

There is no remainder, so the quotient is \( 5x - 2 \).

### Conclusion:
The correct answer is **\( 5x - 2 \)**, which corresponds to the first option.

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Would you like a further breakdown of these steps or additional details?

Here are five related questions to explore further:
1. How does polynomial long division differ from synthetic division?
2. What happens if the remainder is non-zero in polynomial division?
3. Can you use the division algorithm for polynomials to verify the result?
4. How would this process change if the divisor had a higher degree?
5. What role do coefficients play in simplifying the division process?

**Tip:** Always align terms by their degrees carefully when performing polynomial long division to avoid errors.

Use long division to divide the polynomial 15x^2 + 14x - 8 by 3x + 4. What is the quotient?

Learn how to perform polynomial long division by dividing 15x^2 + 14x - 8 by 3x + 4. The step-by-step solution includes finding the quotient and remainder, and is suitable for high school students in Grades 10-12.

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