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The question is asking for a factor of the polynomial \(2x^3 - 3x^2 - 3x + 2\), given that one factor is \((x - 2)\). We need to find which expression represents the other factor or factors of the polynomial.

Let's solve the problem step by step using polynomial division:

### Step 1: Perform Polynomial Division

We will divide \(2x^3 - 3x^2 - 3x + 2\) by \(x - 2\).

#### Division Process:
\[
\frac{2x^3 - 3x^2 - 3x + 2}{x - 2}
\]
1. Divide \(2x^3\) by \(x\): The result is \(2x^2\).
2. Multiply \(2x^2\) by \((x - 2)\), which gives \(2x^3 - 4x^2\).
3. Subtract \((2x^3 - 4x^2)\) from the original polynomial:
\[
(2x^3 - 3x^2 - 3x + 2) - (2x^3 - 4x^2) = x^2 - 3x + 2
\]
4. Divide \(x^2\) by \(x\): The result is \(x\).
5. Multiply \(x\) by \((x - 2)\), which gives \(x^2 - 2x\).
6. Subtract \((x^2 - 2x)\) from the result:
\[
(x^2 - 3x + 2) - (x^2 - 2x) = -x + 2
\]
7. Divide \(-x\) by \(x\): The result is \(-1\).
8. Multiply \(-1\) by \((x - 2)\), which gives \(-x + 2\).
9. Subtract \((-x + 2)\) from the result:
\[
(-x + 2) - (-x + 2) = 0
\]

Thus, the quotient is \(2x^2 + x - 1\).

### Step 2: Factor the Quotient \(2x^2 + x - 1\)

We need to factor \(2x^2 + x - 1\).

1. Find two numbers that multiply to \(2 \times (-1) = -2\) and add to \(1\). These numbers are \(2\) and \(-1\).
2. Rewrite the middle term:
\[
2x^2 + 2x - x - 1
\]
3. Factor by grouping:
\[
2x(x + 1) - 1(x + 1) = (2x - 1)(x + 1)
\]

### Step 3: Final Factorization

The complete factorization of the polynomial \(2x^3 - 3x^2 - 3x + 2\) is:
\[
(x - 2)(2x - 1)(x + 1)
\]

### Step 4: Select the Correct Answer

Looking at the options:
- Option B: \((2x - 1)(x + 1)\) matches the correct factorization.

Thus, the correct answer is **B**.

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Do you have any questions or would you like more details?

Here are 5 related questions to expand your understanding:
1. How would you factor a quartic polynomial?
2. What is synthetic division, and when is it used?
3. Can all cubic polynomials be factored using rational numbers?
4. How does the Remainder Theorem help in polynomial division?
5. What are the steps to identify the roots of a polynomial from its factors?

**Tip**: When factoring higher-degree polynomials, always check if you can use the Rational Root Theorem to find possible roots.

One factor of the polynomial 2x^3 - 3x^2 - 3x + 2 is (x - 2). Which expression represents the other factor, or factors, of the polynomial?

Learn how to factor the cubic polynomial 2x^3 - 3x^2 - 3x + 2, given one factor (x - 2). This problem involves polynomial division and factoring quadratics, key concepts in algebra for students in Grades 10-12.

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