`x^2ltxlt1`
`{:("Quantity A","Quantity B"),((x^(3)-x)(4x+3),(x^(2)+1)(4x^(2)+3x)):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the problem, we need to compare two quantities given the condition \( x^2 < x < 1 \). Let's break down the solution step by step.

### Step 1: Understand the condition
The condition \( x^2 < x < 1 \) implies that \( x \) is a positive number less than 1. Since \( x^2 < x \), we can also deduce that \( x \) must be between 0 and 1.

### Step 2: Define the quantities
We have two quantities to compare:
- Quantity A: \( (x^3 - x)(4x + 3) \)
- Quantity B: \( (x^2 + 1)(4x^2 + 3x) \)

### Step 3: Simplify Quantity A
Let's simplify Quantity A:
\[ x^3 - x = x(x^2 - 1) = x(x - 1)(x + 1) \]
Thus, we can rewrite Quantity A as:
\[ (x(x - 1)(x + 1))(4x + 3) \]

### Step 4: Simplify Quantity B
Now, let's simplify Quantity B:
\[ (x^2 + 1)(4x^2 + 3x) \]
This expression is already simplified.

### Step 5: Compare the two quantities
Now we need to compare:
- Quantity A: \( x(x - 1)(x + 1)(4x + 3) \)
- Quantity B: \( (x^2 + 1)(4x^2 + 3x) \)

### Step 6: Factor out common terms
Notice that both quantities have a common factor of \( 4x^2 + 3x \). However, we need to analyze the first factors:
- For Quantity A, the first factor is \( x(x - 1)(x + 1) \).
- For Quantity B, the first factor is \( x^2 + 1 \).

### Step 7: Analyze the first factors
1. **For Quantity A**: Since \( 0 < x < 1 \), we know:
- \( x - 1 < 0 \) (negative)
- \( x + 1 > 0 \) (positive)
- Therefore, \( x(x - 1)(x + 1) < 0 \) (since it is a product of a positive and a negative number).

2. **For Quantity B**: The first factor \( x^2 + 1 > 0 \) for all \( x \), and since \( x^2 \) is always non-negative, \( x^2 + 1 \) is always positive.

### Step 8: Conclusion
Since Quantity A is negative and Quantity B is positive, we can conclude that:
\[ \text{Quantity B} > \text{Quantity A} \]

Thus, the answer is that Quantity B is greater than Quantity A.

### Final Answer
**Quantity B is greater than Quantity A.**