Example  4
Simplify the expression \(\frac{{a^2 - 3a - 18}}{{a^2 + 5a + 6}}\). Identify the non-permissible values.


Step 1: Factor each polynomial.

\[\frac{{a^2 - 3a - 18}}{{a^2 + 5a + 6}} = \frac{(a - 6)\left( {a + 3} \right)}{{\left( {a + 2} \right)\left( {a + 3} \right)}}\]

Step 2: Determine the NPVs.

\(\begin{align}
 a + 2 &\ne 0 \\
 a &\ne -2 \\
\end{align}\)
\(\begin{align}
 a + 3 &\ne 0 \\
 a &\ne -3 \\
 \end{align}\)

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

\[\frac{{(a - 6) {\color{red} \cancel {\color{#444} {\left( {a + 3} \right)}}}}}{{\left( {a + 2} \right)  {\color{red} \cancel {\color{#444} {\left( {a + 3} \right)}}}}} = \frac{{a - 6}}{{a + 2}}\]

The simplified expression is \(\frac{{a - 6}}{{a + 2}}, \thinspace a \ne - 2, - 3\)