Example  3
Simplify the expression \(\frac{{27m^2n + 18mn}}{{3m^2 + 12m}}\).  Identify the non-permissible values.


Step 1: Factor each polynomial.

\[\frac{{27m^2n + 18mn}}{{3m^2 + 12m}} = \frac{{9mn\left( {3m + 2} \right)}}{{3m\left( {m + 4} \right)}}\]

Step 2: Determine the NPVs.

\(m \ne 0\)
\(\begin{align}
 m + 4 &\ne 0 \\
 m &\ne -4 \\
 \end{align}\)

Step 3: Simplify by removing factors found in both the numerator and denominator.

\[\frac{{9mn \left( {3m + 2} \right)}}{{3m\left({m + 4} \right)}} = \frac{{3n\left( {3m + 2} \right)}}{{m + 4}}\]

The simplified expression is \(\frac{{3n\left( {3m + 2} \right)}}{{m + 4}},\thinspace m \ne 0, -4\).