Question: (\[ \frac{x^2 - 2x - 24}{x^2 - 16} \]\)…

\(\[ \frac{x^2 - 2x - 24}{x^2 - 16} \]\)

Solution

The problem is a fraction with the expression \( \frac{x^2 - 2x - 24}{x^2 - 16} \). First, factor the numerator and the denominator. For the numerator \( x^2 - 2x - 24 \): Find two numbers that multiply to \(-24\) and add to \(-2\). These numbers are \(4\) and \(-6\). \[ x^2 - 2x - 24 = (x - 6)(x + 4) \] For the denominator \( x^2 - 16 \): This is a difference of squares. \[ x^2 - 16 = (x - 4)(x + 4) \] Now, the expression is: \[ \frac{(x - 6)(x + 4)}{(x - 4)(x + 4)} \] Cancel the common factor \((x + 4)\). The simplified expression is: \[ \frac{x - 6}{x - 4} \]