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Factoring polynomials is an essential skill in simplifying rational expressions. To factor a polynomial, we rewrite it as a product of its factors. These factors can include numbers, variables, and expressions. Consider the given problem. The numerator is \(9r^2 - 4s^2\), which is a polynomial. This polynomial can be rewritten in factored form by identifying it as a difference of squares. In general, when we have a polynomial like \(a^2 - b^2\), it can be factored as: \[ (a-b)(a+b). \] For our problem, we first identify \(a = 3r\) and \(b = 2s\). Thus, the factorization of the numerator becomes: \[ 9r^2 - 4s^2 = (3r)^2 - (2s)^2 = (3r - 2s)(3r + 2s). \] It's crucial to get comfortable with recognizing patterns in polynomials to factor them efficiently.

The difference of squares is a common factorization technique used when we have two terms, each a perfect square, separated by a minus sign. In our example, the numerator is \(9r^2 - 4s^2\). To identify a difference of squares: \[ 1. Ensure both terms are perfect squares.\
2. Check for a subtraction sign between them. \] For \(9r^2\), we recognize it as \((3r)^2\). For \(4s^2\), it is \((2s)^2\). So the expression \(9r^2 - 4s^2\) matches the difference of squares form \((a^2 - b^2)\), and can be factored as \((a - b)(a + b)\). Therefore: \[ 9r^2 - 4s^2 = (3r - 2s)(3r + 2s). \] This pattern is fundamental and frequently used in algebra to simplify expressions.

The Greatest Common Factor (GCF) is the largest factor shared by all terms in a polynomial. When simplifying rational expressions, finding and factoring out the GCF is vital. In the denominator of our given problem, \(9r + 6s\), both terms share a common factor of 3. To factor out the GCF: \[ 1. Identify the GCF of all terms in the polynomial.\
2. Divide each term by the GCF and factor it out. \] For \(9r + 6s\), we divide each term by 3: \[ 9r \div 3 = 3r\
6s \div 3 = 2s\. \] We factor out the 3: \[ 9r + 6s = 3(3r + 2s). \] Simplifying rational expressions often involves both factoring polynomials and recognizing common factors in the numerator and denominator. This simplification helps reduce the expression to its lowest terms.