Manipulating Rational Expressions

We want to subtract the given rational expressions. $$\begin{array}{c}\frac{4}{3x+6}-\frac{x+1}{x^2-4}\end{array}$$ To do so, we will start by factoring the denominators. Let's factor the first one by factoring out the $3.$ $$\begin{array}{c}3x+6\stackrel{\text{Factor out }3}{\to }3(x+2)\end{array}$$ Now, let's factor the second denominator. Next, we will rewrite the expression with the factored denominators and determine the least common denominator (LCD). $$\begin{array}{c}\frac{4}{3x+6}-\frac{x+1}{x^2-4}\\ \Updownarrow \\ \frac{4}{3(x+2)}-\frac{x+1}{(x+2)(x-2)}\end{array}$$ The LCD is $3(x+2)(x-2).$ Now, we can subtract the expressions by rewriting each of them using the LCD.