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The given expression is:

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

We need to simplify this expression and compare it to the provided answer choices. Here's the process:

### Step 1: Find the common denominator
The denominators of the two terms are \( x - 1 \) and \( 3x + 2 \). The least common denominator (LCD) is:

\[
(x - 1)(3x + 2)
\]

### Step 2: Rewrite each fraction with the common denominator
For the first fraction \( \frac{3x + 2}{x - 1} \), multiply both the numerator and denominator by \( 3x + 2 \) to get:

\[
\frac{(3x + 2)(3x + 2)}{(x - 1)(3x + 2)} = \frac{(3x + 2)^2}{(x - 1)(3x + 2)}
\]

For the second fraction \( \frac{x + 4}{3x + 2} \), multiply both the numerator and denominator by \( x - 1 \) to get:

\[
\frac{(x + 4)(x - 1)}{(x - 1)(3x + 2)}
\]

### Step 3: Subtract the numerators
Now, subtract the numerators over the common denominator:

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

### Step 4: Expand the terms
First, expand \( (3x + 2)^2 \):

\[
(3x + 2)(3x + 2) = 9x^2 + 12x + 4
\]

Next, expand \( (x + 4)(x - 1) \):

\[
(x + 4)(x - 1) = x^2 + 3x - 4
\]

### Step 5: Perform the subtraction
Now subtract \( x^2 + 3x - 4 \) from \( 9x^2 + 12x + 4 \):

\[
(9x^2 + 12x + 4) - (x^2 + 3x - 4) = 9x^2 + 12x + 4 - x^2 - 3x + 4 = 8x^2 + 9x + 8
\]

### Step 6: Final simplified expression
The simplified expression is:

\[
\frac{8x^2 + 9x + 8}{(x - 1)(3x + 2)}
\]

This matches option **B**.

### Final Answer: **B**

Would you like a further breakdown or any other questions on this topic?

Here are five related questions to expand your understanding:

1. How do you find the least common denominator of two rational expressions?
2. How do you expand a binomial expression like \( (a + b)^2 \)?
3. What are the key steps to subtracting fractions with different denominators?
4. How do you simplify complex rational expressions involving quadratic terms?
5. What is the significance of factoring when dealing with rational expressions?

**Tip:** Always check your work by expanding the factored forms to ensure you've simplified correctly.

Which expression is equivalent to \( \frac{3x + 2}{x - 1} - \frac{x + 4}{3x + 2} \)?

Learn how to subtract the rational expressions \( \frac{3x + 2}{x - 1} - \frac{x + 4}{3x + 2} \) using the least common denominator method. This step-by-step solution expands binomials and simplifies quadratic expressions, perfect for Grades 10-12 students.

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