Let's simplify the given expression step by step:

\(

\frac{x + 2}{x + 1} - \frac{x - 2}{x + 2}

\)

To subtract the fractions, we need to find a common denominator, which in this case is \((x + 1)(x + 2)\).

First, let's rewrite the fractions with the common denominator:

\(

\frac{(x + 2)(x + 2)}{(x + 1)(x + 2)} - \frac{(x - 2)(x + 1)}{(x + 1)(x + 2)}

\)

Now, expand the numerators:

\(

\frac{x^2 + 4x + 4}{(x + 1)(x + 2)} - \frac{x^2 - x - 2}{(x + 1)(x + 2)}

\)

Now, we can subtract the fractions:

\(

\frac{(x^2 + 4x + 4) - (x^2 - x - 2)}{(x + 1)(x + 2)}

\)

Simplify the numerators:

\(

\frac{x^2 + 4x + 4 - x^2 + x + 2}{(x + 1)(x + 2)}

\)

Combine like terms in the numerator:

\(

\frac{5x + 6}{(x + 1)(x + 2)}

\)

Therefore, the simplified expression is:

\(

\frac{5x + 6}{(x + 1)(x + 2)}

\)

This matches option C: \(\frac{5x + 6}{(x + 1)(x + 2)}\).