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

Answer: (D)

Verified step by step guidance

1. Identify the given rational expressions: \(\frac{x+2}{x^2 - 4} - \frac{1}{x - 2}\).
2. Factor the denominator \(x^2 - 4\) as a difference of squares: \(x^2 - 4 = (x - 2)(x + 2)\).
3. Rewrite the first fraction with the factored denominator: \(\frac{x+2}{(x - 2)(x + 2)}\).
4. Express the second fraction with the common denominator \((x - 2)(x + 2)\) by multiplying numerator and denominator by \((x + 2)\): \(\frac{1}{x - 2} = \frac{x + 2}{(x - 2)(x + 2)}\).
5. Subtract the numerators over the common denominator: \(\frac{x + 2}{(x - 2)(x + 2)} - \frac{x + 2}{(x - 2)(x + 2)} = \frac{(x + 2) - (x + 2)}{(x - 2)(x + 2)}\).