Solve the inequality
$$ \frac{x^4 + x^2 + 1}{x^2 - 4x - 5}

Question

Solve the inequality  
$$ \frac{x^4 + x^2 + 1}{x^2 - 4x - 5} < 0 $$

Accepted Answer

To solve the inequality $$ \frac{x^4 + x^2 + 1}{x^2 - 4x - 5} < 0 $$, follow these steps:

**Step 1: Analyze the numerator and denominator**  
The numerator is $$ x^4 + x^2 + 1 $$, which is always positive because all terms are non-negative for any real value of $$ x $$.  
The denominator is $$ x^2 - 4x - 5 $$. Factorize the denominator:  
$$
x^2 - 4x - 5 = (x - 5)(x + 1)
$$

**Step 2: Determine the critical points**  
The critical points occur where the denominator equals zero:  
$$
x = -1 \quad 	ext{and} \quad x = 5
$$

**Step 3: Test intervals based on critical points**  
The sign of the rational expression depends on the intervals formed by the critical points:  
- Interval $$ (-\infty, -1) $$
- Interval $$ (-1, 5) $$
- Interval $$ (5, \infty) $$

Since the numerator is always positive, the sign of the expression depends solely on the denominator:
- In $$ (-\infty, -1) $$: The denominator is positive.
- In $$ (-1, 5) $$: The denominator is negative.
- In $$ (5, \infty) $$: The denominator is positive.

Thus, the inequality holds in the interval where the denominator is negative:  
$$
-1 < x < 5
$$

**Step 4: Verify options**  
- Option (a): Incorrect because it includes regions outside of the valid interval.  
- Option (b): Correct because it matches the solution interval.  
- Option (c): Incorrect because it only includes values greater than 5.  
- Option (d): Incorrect because it includes values less than -1.

Thus, the correct answer is **b. $$ -1 < x < 5 $$**.

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