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The problem asks to determine the quotient of the division:

\[
\frac{2x^3 + 4x - 5x^2 + 3}{2x - 1}
\]

We can solve this by performing polynomial long division.

### Step 1: Arrange the terms in descending order of powers of \( x \)

\[
\frac{2x^3 - 5x^2 + 4x + 3}{2x - 1}
\]

### Step 2: Start dividing

- Divide the first term of the numerator \( 2x^3 \) by the first term of the denominator \( 2x \), which gives \( x^2 \).
  
- Multiply \( x^2 \) by \( 2x - 1 \), which gives \( 2x^3 - x^2 \).

- Subtract \( (2x^3 - x^2) \) from \( 2x^3 - 5x^2 + 4x + 3 \):
  \[
  (2x^3 - 5x^2 + 4x + 3) - (2x^3 - x^2) = -4x^2 + 4x + 3
  \]

### Step 3: Repeat the process

- Divide \( -4x^2 \) by \( 2x \), which gives \( -2x \).
  
- Multiply \( -2x \) by \( 2x - 1 \), which gives \( -4x^2 + 2x \).

- Subtract \( (-4x^2 + 2x) \) from \( -4x^2 + 4x + 3 \):
  \[
  (-4x^2 + 4x + 3) - (-4x^2 + 2x) = 2x + 3
  \]

### Step 4: Continue

- Divide \( 2x \) by \( 2x \), which gives \( 1 \).

- Multiply \( 1 \) by \( 2x - 1 \), which gives \( 2x - 1 \).

- Subtract \( (2x - 1) \) from \( 2x + 3 \):
  \[
  (2x + 3) - (2x - 1) = 4
  \]

### Step 5: Conclusion

The quotient is:

\[
q(x) = x^2 - 2x + 1
\]

So the correct answer is **C)** \( q(x) = x^2 - 2x + 1 \).

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Would you like any more details or have any questions?

Here are some related questions you might find helpful:
1. How do you handle polynomial division when the divisor has more than two terms?
2. What are the conditions for synthetic division to be used instead of long division?
3. How do you verify the result of a polynomial division?
4. Can remainders be written as fractional terms in polynomial division?
5. How does the degree of the quotient compare to the degree of the original polynomial?

**Tip:** Always arrange terms in descending powers of the variable to avoid mistakes during polynomial division.

Determine el cociente de la siguiente división: (2x^3 + 4x - 5x^2 + 3) / (2x - 1).

Learn how to determine the quotient for the division of the polynomial (2x^3 + 4x - 5x^2 + 3) by (2x - 1). This detailed step-by-step solution covers polynomial long division and algebraic concepts, suitable for high school students in Grades 9-12.

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