Manipulating Rational Expressions

We want to subtract the given rational expressions. $$\begin{array}{c}\frac{3a+2}{a^2-16}-\frac{7}{6a+24}\end{array}$$ To do so, we will start by factoring the denominators. Let's factor the first one. Now, let's factor the second denominator by factoring out the $6.$ $$\begin{array}{c}6a+24\stackrel{\text{Factor out }6}{\to }6(a+4)\end{array}$$ Next, we will rewrite the expression with the factored denominators and determine the least common denominator (LCD). $$\begin{array}{c}\frac{3a+2}{a^2-16}-\frac{7}{6a+24}\\ \Updownarrow \\ \frac{3a+2}{(a+4)(a-4)}-\frac{7}{6(a+4)}\end{array}$$ The LCD is $6(a+4)(a-4).$ We can subtract and add the expressions by rewriting each of them using the LCD.