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: \(x^2 - 4 = (x - 2)(x + 2)\), so rewrite the first fraction as \(\frac{x+2}{(x-2)(x+2)}\).
3. Recognize that the second fraction \(\frac{1}{x-2}\) has a denominator missing the factor \((x+2)\), so find a common denominator for both fractions, which is \((x-2)(x+2)\).
4. Rewrite the second fraction with the common denominator by multiplying numerator and denominator by \((x+2)\): \(\frac{1}{x-2} = \frac{x+2}{(x-2)(x+2)}\).
5. Now subtract the two fractions with the common denominator: \(\frac{x+2}{(x-2)(x+2)} - \frac{x+2}{(x-2)(x+2)}\), combine the numerators over the common denominator, and simplify the numerator to find the difference.