Question

simplifying rational functions
\\(\frac{x^2 - 2x - 3}{x^2 + 6x + 5}\\)

Explanation:

Step1: Factor numerator and denominator

Factor \(x^2 - 2x - 3\): find two numbers that multiply to \(-3\) and add to \(-2\), which are \(-3\) and \(1\). So \(x^2 - 2x - 3=(x - 3)(x + 1)\).
Factor \(x^2 + 6x + 5\): find two numbers that multiply to \(5\) and add to \(6\), which are \(5\) and \(1\). So \(x^2 + 6x + 5=(x + 5)(x + 1)\).

Step2: Cancel common factors

The rational function becomes \(\frac{(x - 3)(x + 1)}{(x + 5)(x + 1)}\). Cancel the common factor \((x + 1)\) (where \(x
eq - 1\) to avoid division by zero).

Answer:

\(\frac{x - 3}{x + 5}\) (for \(x
eq - 1, - 5\))