Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{x+2}{x^2-4}-\frac{1}{x-2}$

• A. $\frac{1}{(x-2)(x+2)}$
• B. $\frac{x+2}{(x-2)(x+2)}$
• C. $\frac{x-2}{(x-2)(x+2)}$
• D. $0$

Answer: (D)

Verified step by step guidance

1. Identify the denominators of the rational expressions: the first denominator is \(x^2 - 4\) and the second denominator is \(x - 2\).
2. Factor the denominator \(x^2 - 4\) as a difference of squares: \(x^2 - 4 = (x - 2)(x + 2)\).
3. Rewrite both rational expressions with the common denominator \((x - 2)(x + 2)\): the first fraction remains \(\frac{x+2}{(x-2)(x+2)}\), and the second fraction becomes \(\frac{1}{x-2} = \frac{x+2}{(x-2)(x+2)}\) after multiplying numerator and denominator by \((x+2)\).
4. Set up the subtraction with the common denominator: \(\frac{x+2}{(x-2)(x+2)} - \frac{x+2}{(x-2)(x+2)}\).
5. Subtract the numerators over the common denominator: \(\frac{(x+2) - (x+2)}{(x-2)(x+2)}\), then simplify the numerator and denominator if possible.