The solution set of `((x^(2)+x+1)(x^(2))log(x))/((x+1)(x+9))lt 0` is

Options

A. `(0,1)uu(1,2)`
B. `(1,2)`
C. `(0,1)`
D. `(0,2)`

Explanation

To solve the inequality

\[ \frac{(x^2 + x + 1)(x^2 \log x)}{(x + 1)(x + 9)} < 0, \]

we will follow these steps:

### Step 1: Identify the critical points
The critical points occur when the numerator or denominator is zero.

1. **Numerator**:
- \(x^2 + x + 1 = 0\) has no real roots since its discriminant \(B^2 - 4AC = 1^2 - 4(1)(1) = 1 - 4 = -3 < 0\). Thus, \(x^2 + x + 1 > 0\) for all \(x\).
- \(x^2 \log x = 0\) gives \(x = 0\) or \(\log x = 0\) which gives \(x = 1\).

2. **Denominator**:
- \(x + 1 = 0\) gives \(x = -1\).
- \(x + 9 = 0\) gives \(x = -9\).

Thus, the critical points are \(x = 0\), \(x = 1\), \(x = -1\), and \(x = -9\).

### Step 2: Determine the intervals
The critical points divide the number line into intervals. We will analyze the sign of the expression in each interval. The intervals are:

1. \( (-\infty, -9) \)
2. \( (-9, -1) \)
3. \( (-1, 0) \)
4. \( (0, 1) \)
5. \( (1, \infty) \)

### Step 3: Test the sign of the expression in each interval
We will check the sign of the expression in each interval.

1. **Interval \( (-\infty, -9) \)**: Choose \(x = -10\)
- The numerator is positive (since \(x^2 + x + 1 > 0\) and \(x^2 \log x\) is not defined).
- The denominator is negative. Thus, the expression is undefined.

2. **Interval \( (-9, -1) \)**: Choose \(x = -5\)
- The numerator is positive.
- The denominator is negative. Thus, the expression is negative.

3. **Interval \( (-1, 0) \)**: Choose \(x = -0.5\)
- The numerator is positive.
- The denominator is negative. Thus, the expression is negative.

4. **Interval \( (0, 1) \)**: Choose \(x = 0.5\)
- The numerator is positive (since \(x^2 + x + 1 > 0\) and \(x^2 \log x < 0\)).
- The denominator is positive. Thus, the expression is negative.

5. **Interval \( (1, \infty) \)**: Choose \(x = 2\)
- The numerator is positive.
- The denominator is positive. Thus, the expression is positive.

### Step 4: Compile the results
From the testing, we find that the expression is negative in the intervals:

- \( (-9, -1) \)
- \( (-1, 0) \)
- \( (0, 1) \)

However, since the logarithm is only defined for \(x > 0\), we discard the intervals that include negative values.

### Final Solution
The solution set for the inequality

\[ \frac{(x^2 + x + 1)(x^2 \log x)}{(x + 1)(x + 9)} < 0 \]

is

\[ (0, 1). \]