Question

1. \\(\frac{1}{x^2 - 16} \cdot \frac{x^2 - x - 12}{x^2 + 5x - 6}\\)

Explanation:

Step1: Factor each quadratic

Factor \(x^2 - 16\) (difference of squares): \(x^2 - 16=(x - 4)(x + 4)\)
Factor \(x^2 - x - 12\) (find two numbers: \(-4\) and \(3\)): \(x^2 - x - 12=(x - 4)(x + 3)\)
Factor \(x^2 + 5x - 6\) (find two numbers: \(6\) and \(-1\)): \(x^2 + 5x - 6=(x + 6)(x - 1)\)

Step2: Substitute factored forms

Substitute into the expression:
\(\frac{1}{(x - 4)(x + 4)} \cdot \frac{(x - 4)(x + 3)}{(x + 6)(x - 1)}\)

Step3: Cancel common factors

Cancel \((x - 4)\) from numerator and denominator:
\(\frac{1}{(x + 4)} \cdot \frac{(x + 3)}{(x + 6)(x - 1)}\)

Step4: Multiply numerators and denominators

Multiply remaining terms:
\(\frac{x + 3}{(x + 4)(x + 6)(x - 1)}\)

Answer:

\(\boldsymbol{\frac{x + 3}{(x + 4)(x + 6)(x - 1)}}\) (or expanded denominator: \(\frac{x + 3}{x^3 + 9x^2 + 14x - 24}\))