Simplify the complex fraction. Write your answer in factored form. \( \frac{\frac{1}{3 x^{2}-3}}{\frac{5}{x+1}-\frac{x+4}{x^{2}-3 x-4}}=\square, x \neq-1, x \neq 4 \)

To simplify the given complex fraction, we start by simplifying the denominator: 1. The expression in the denominator is \( \frac{5}{x+1} - \frac{x+4}{x^2 - 3x - 4} \). 2. First, we factor the quadratic \( x^2 - 3x - 4 \). It factors to \( (x - 4)(x + 1) \). 3. Rewrite the fraction: \[ \frac{5}{x+1} - \frac{x+4}{(x-4)(x+1)} \] 4. To combine these fractions, find a common denominator, which is \( (x - 4)(x + 1) \): \[ \frac{5(x-4)}{(x-4)(x+1)} - \frac{x+4}{(x-4)(x+1)} \] 5. Combine the fractions: \[ \frac{5(x-4) - (x+4)}{(x-4)(x+1)} = \frac{5x - 20 - x - 4}{(x - 4)(x + 1)} = \frac{4x - 24}{(x - 4)(x + 1)} = \frac{4(x - 6)}{(x - 4)(x + 1)} \] 6. We can now substitute this back into the original complex fraction: \[ \frac{\frac{1}{3x^2 - 3}}{\frac{4(x-6)}{(x-4)(x+1)}} \] 7. Simplify the fraction by multiplying by the reciprocal of the denominator: \[ \frac{1}{3(x^2 - 1)} \cdot \frac{(x - 4)(x + 1)}{4(x - 6)} \] 8. The term \( 3(x^2 - 1) \) factors to \( 3(x - 1)(x + 1) \): \[ \frac{(x - 4)(x + 1)}{4 \cdot 3(x - 1)(x + 1)(x - 6)} \] 9. Cancel the \( (x + 1) \) terms: \[ = \frac{(x - 4)}{12(x - 1)(x - 6)} \] Thus, we have simplified the complex fraction to: \[ \frac{x - 4}{12(x - 1)(x - 6)} \] In factored form, the final answer is: \[ \frac{x - 4}{12(x - 1)(x - 6)} \]