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Understanding how to factor quadratics is essential for simplifying algebraic expressions. A quadratic expression typically has the form \(ax^2 + bx + c\), where \((a\), \((b\), and \((c\) are constants. Factoring involves converting this expression into a product of two binomials. The goal is to find two numbers that multiply to \((a\times c\) and add up to \((b\), and then to use these numbers to create the binomials.

For example, in our exercise, the quadratic expression \((x^2 + x - 20\) was factored into \( (x - 4)(x + 5) \). The numbers that multiply to \( -20 \) and add up to \((1\) are \( -4 \) and \( 5 \), hence the binomials. Factoring is a critical step before you can simplify the expression further. It's important to practice recognizing patterns and applying them to factoring different quadratics.

Once we have factored the numerators and denominators in a rational expression, we should look for common factors that appear in both. Canceling common factors is a method that simplifies the expression, much like reducing fractions in basic arithmetic.

For instance, after factoring the example expression, we identified \( (x+5) \) as a common factor in both the numerator and the denominator. By the division property of equality, a factor that appears in both the top and bottom can be eliminated, simplifying the expression to \( \frac{x-4}{x-3} \). This process is akin to simplifying the fraction \( \frac{15}{30} \) to \( \frac{1}{2} \), where \( 15 \) is the common factor. It's important not to cancel out terms that are not common factors as it can lead to incorrect results.

To simplify rational expressions, one needs to factorize the numerator and the denominator, then reduce the expression by canceling out common factors. A rational expression is much like a fraction but with polynomials in the numerator and denominator.

The simplification process requires careful factoring to ensure no common factors are missed. As seen in the given example, once we factor and cancel the common factor \( (x+5) \), we arrive at a simpler form of the rational expression: \( \frac{x-4}{x-3} \). Remember that rational expressions cannot be simplified if there are no common factors after factoring. Also, one must always check to ensure that the variable does not take on values that would make the denominator zero, as this would result in an undefined expression.