Example  4
Simplify the expression \(\frac{{2x + 5}}{{2x^2 + 9x + 4}} + \frac{3}{{2x + 1}}\).  Identify any non-permissible values.


Step 1: Identify the NPVs.

\(2x^2 + 9x + 4 = \left( {2x + 1} \right)\left( {x + 4} \right)\)

\(\begin{align}
 2x + 1 &\ne 0 \\
 x &\ne -\frac{1}{2} \\
 \end{align}\)
\(\begin{align}
 x + 4 &\ne 0 \\
 x &\ne -4 \\
 \end{align}\)

Step 2: Determine the LCD.

The LCD is \(\left( {2x + 1} \right)\left( {x + 4} \right)\).

Step 3: Use the LCD to rewrite each term as an equivalent rational expression with the same denominator, and simplify.

\[\begin{align}
 \frac{{2x + 5}}{{2x^2  + 9x + 4}} + \frac{3}{{2x + 1}} &= \frac{{2x + 5}}{{\left( {2x + 1} \right)\left( {x + 4} \right)}} + \frac{{3\left( {\color{red}{x + 4}} \right)}}{{\left( {2x + 1} \right)\left( {\color{red}{x + 4}} \right)}} \\
  &= \frac{{2x + 5}}{{\left( {2x + 1} \right)\left( {x + 4} \right)}} + \frac{{3\left( {x + 4} \right)}}{{\left( {2x + 1} \right)\left( {x + 4} \right)}} \\
  &= \frac{{2x + 5 + 3x + 12}}{{\left( {2x + 1} \right)\left( {x + 4} \right)}} \\
  &= \frac{{5x + 17}}{{\left( {2x + 1} \right)\left( {x + 4} \right)}},\thinspace x \ne - 4, \thinspace -\frac{1}{2} \\
 \end{align}\]