Example  4
Simplify the rational expression \(\frac{{x^2 - 9}}{{6x^2 + 7x + 2}} \cdot \frac{{2x^2 - 5x - 3}}{{3x^2 + 11x + 6}}\). Identify any non-permissible values.


Step 1: Factor all polynomials. (Do not simplify yet!)


\[\frac{{x^2 - 9}}{{6x^2 + 7x + 2}} \cdot \frac{{2x^2 - 5x - 3}}{{3x^2 + 11x + 6}} = \frac{{\left( {x - 3} \right)\left( {x + 3} \right)}}{{\left( {2x + 1} \right)\left( {3x + 2} \right)}} \cdot \frac{{\left( {2x + 1} \right)\left( {x - 3} \right)}}{{\left( {x + 3} \right)\left( {3x + 2} \right)}}\]

Step 2: Identify the NPVs.

\[\begin{align}
 2x + 1 &\ne 0 \\
 x &\ne -\frac{1}{2} \\
 \end{align}\]
\[\begin{align}
 3x + 2 &\ne 0 \\
 x &\ne -\frac{2}{3} \\
 \end{align}\]
\[\begin{align}
 x + 3 &\ne 0 \\
 x &\ne -3 \\
 \end{align}\]

Step 3: Simplify the expression.

\[\begin{align}
 \frac{{x^2 - 9}}{{6x^2 + 7x + 2}} \cdot \frac{{2x^2 - 5x - 3}}{{3x^2 + 11x + 6}} &= \frac{{\left( {x - 3} \right) {\color{red}\cancel {\color{#444}{\left(x + 3 \right)}}}}}{{{\color{red}\cancel {\color{#444}{\left(2x + 1 \right)}}}\left( {3x + 2} \right)}} \cdot \frac{{{\color{red} \cancel {\color{#444} {\left(2x + 1 \right)}}}\left( {x - 3} \right)}}{{{\color{red}\cancel {\color{#444}{\left(x + 3 \right)}}}\left( {3x + 2} \right)}} \\
  &= \frac{{x - 3}}{{3x + 2}} \cdot \frac{{x - 3}}{{3x + 2}} \\
  &= \frac{{\left( {x - 3} \right)^2 }}{{\left( {3x + 2} \right)^2 }},\thinspace x \ne - 3, -\frac{2}{3}, -\frac{1}{2} \\
 \end{align}\]