Explanation
Simplification of rational expressions involves reducing a rational expression to its simplest form. A rational expression is a fraction where both the numerator and the denominator are polynomials. The goal of simplification is to make this fraction as simple as possible.
Steps to Simplify Rational Expressions

Factor both the numerator and the denominator:
Identify and factor out the greatest common factors (GCF) of the polynomials in both the numerator and the denominator.

Cancel out common factors:
Any factor that appears in both the numerator and the denominator can be cancelled out.
Example
Consider the rational expression:
$\frac{6x^2 + 12x}{3x}$
Step 1: Factor the numerator and the denominator.
 Numerator: $6x^2 + 12x = 6x(x + 2)$
 Denominator: $3x = 3x$
So the expression becomes:
$\frac{6x(x + 2)}{3x}$
Step 2: Cancel out the common factors between the numerator and the denominator.
Since $6x$ and $3x$ share a common factor of $3x$, we cancel it out:
$\frac{6x(x + 2)}{3x} = \frac{2(x + 2)}{1} = 2(x + 2)$
So, the simplified form is:
$2(x + 2)$
Important Notes
 Only common factors can be cancelled out.
 Ensure that the rational expression is defined (denominator ≠ 0) after simplifying.
 Simplification does not change the value but makes it easier to understand or use in further calculations.
Common Mistakes
 Leaving the factorization incomplete: Make sure to fully factorize both the numerator and the denominator.
 Cancelling terms that are not factors: Only factors, not terms, can be cancelled. For example, in $\frac{x+2}{x+3}$, $x$ cannot be cancelled as they are terms, not factors.
Understanding these fundamental steps and practicing multiple examples can significantly help in mastering the simplification of rational expressions.