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Factoring is a technique we use to simplify complicated algebraic expressions. Think of it like breaking up a big problem into smaller, more manageable parts. In our exercise, we started by factoring the numerator and denominator separately.
To factor an expression, look for the greatest number or term that is common in all the terms of the expression. For example, in the numerator \(3m^2 + 3m\), we noted that both terms contain \(3m\). So, we factored out \(3m\) from each term, getting \(3m(m + 1)\). Similarly, for the denominator \(6m^2 + 9m\), the common factor is \(3m\), and factoring it out gives \(3m(2m + 3)\). Once everything is factored out, the fraction looks simpler and cleaner.

Identifying the common factor is crucial in factoring. When dealing with polynomials, a common factor is a term that can divide each term of the expression without leaving a remainder.
In our exercise, we found that both the numerator (\(3m^2 + 3m\)) and the denominator (\(6m^2 + 9m\)) have \(3m\) as a common factor. By pulling out this common factor, we break down complex expressions into simpler forms. This makes the next steps of simplification much easier.
Always look for the largest common factor to ensure you are simplifying as much as possible. For instance, instead of just pulling out \(m\), pulling out \(3m\) helps to reduce the expression significantly.

Canceling terms is the final step in simplifying algebraic fractions. After factoring, the fraction often has similar terms in the numerator and the denominator. These terms can be canceled out because they are like dividing a number by itself, which always equals 1.
In our exercise, both the numerator and the denominator had a common factor of \(3m\), so we canceled \(3m\) out from both, leaving us with \(\frac{m + 1}{2m + 3}\).
Always remember: You can only cancel terms if they appear in both the numerator and the denominator. After canceling, make sure to re-check the remaining terms to ensure nothing else can be simplified. This will give you the most simplified form of the fraction.