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To simplify rational expressions, the first step involves factoring the numerator and the denominator. For instance, consider the polynomial in the numerator, \(x^2 + 2x - 15\). To factor it, we need to find two numbers that multiply to -15 and add to 2. These numbers are 5 and -3. Thus, the polynomial can be factored as \((x + 5)(x - 3)\).

Similarly, the denominator polynomial, \(x^2 + 6x + 5\), needs to be factored. We look for two numbers that multiply to 5 and add to 6, which are 1 and 5. Therefore, the denominator is factored as \((x + 1)(x + 5)\).

Factoring polynomials is essential to simplification because it reveals common factors between the numerator and the denominator.

Once both the numerator and denominator are factored, the next step is to cancel out any common factors. Common factors are terms that appear in both the numerator and the denominator. In our example, after factoring, the rational expression is:

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

Here, \(x + 5\) is a common factor in both the numerator and the denominator. To simplify the expression, we cancel out \(x + 5\):

\(\frac{x - 3}{x + 1}\).

This step is critical because it reduces the complexity of the expression and brings us closer to the lowest terms.

The goal of simplifying rational expressions is to reduce them to their lowest terms. An expression is in its lowest terms when no further reduction is possible, meaning all common factors have been canceled. Returning to our simplified expression:

\(\frac{x - 3}{x + 1}\)

There are no common factors left between the numerator and the denominator. Thus, \(\frac{x - 3}{x + 1}\) is in its lowest terms. This makes the expression simpler and often easier to work with in further calculations or problem-solving.

Simplifying to lowest terms ensures that the expression is as concise and straightforward as possible, making it easier to interpret and use in further mathematical operations.