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Factoring quadratics is a vital skill in algebra. A quadratic expression is typically written in the form \(ax^2 + bx + c\). To factor it, we look for two numbers that multiply to give the product of \(a\) and \(c\) and add up to \(b\). Let's use the numerator from the problem, \(x^2 + 2x - 15\). Here, \(a\) is 1, \(b\) is 2, and \(c\) is -15.
To factor this expression, we need two numbers that multiply to -15 (\(a * c\)) and add to 2 (\(b\)). These numbers are 5 and -3, so we rewrite the quadratic as \((x+5)(x-3)\).
Practicing this with different quadratic expressions helps you get better at recognizing these pairs quickly. The same steps apply to the denominator, where we factor \(x^2 + 6x + 5\) to \((x+5)(x+1)\).

Following these steps will allow you to factor any quadratic polynomial, an essential step in many algebra problems.

Simplifying fractions involves rewriting them in their simplest form. Once you have factored both the numerator and the denominator of a rational expression, you can start simplifying it.

Let's take our example: \( \frac{(x + 5)(x - 3)}{(x + 5)(x + 1)} \).
Notice that both the numerator (top part) and the denominator (bottom part) share a common factor: \((x + 5)\).

When fractions have common factors, you can cancel them out. Think of common factors as numbers or expressions that divide both parts of the fraction evenly. So cancel out \((x + 5)\) from both the numerator and the denominator, simplifying our fraction to \( \frac{(x - 3)}{(x + 1)} \).

Using this technique ensures fractions are in their simplest form, making them easier to work with in further algebraic operations or solving equations.

Writing a rational expression in its lowest terms means simplifying it as much as possible. This process ensures the fraction cannot be reduced any further.

For our example, after factoring and simplifying, we reach \( \frac{(x - 3)}{(x + 1)} \). There's no common factor left in the numerator and denominator.

This means the fraction is in its lowest terms. No further reduction is possible. To check your work, look for any common factors after simplification.

If there are none, you've successfully written the expression in its lowest terms.

Practice is key to mastering this skill. Working with different rational expressions will help you quickly identify common factors and ensure all your fractions are in their simplest form.