To determine the difference between the given fractions and to match it with the provided options, let's follow a step-by-step method.
Given:
[tex]\[ \frac{x+5}{x+2} - \frac{x+1}{x^2 + 2x} \][/tex]
First, recognize that [tex]\(x^2 + 2x\)[/tex] can be factored:
[tex]\[ x^2 + 2x = x(x + 2) \][/tex]
Rewrite the expression with a common denominator:
[tex]\[ \frac{x+5}{x+2} - \frac{x+1}{x(x+2)} \][/tex]
The common denominator for both fractions is [tex]\(x(x + 2)\)[/tex].
Step 1: Rewrite the first fraction with the common denominator:
[tex]\[ \frac{x+5}{x+2} = \frac{(x+5) \cdot x}{(x+2) \cdot x} = \frac{x(x+5)}{x(x+2)} \][/tex]
Step 2: The second fraction remains the same:
[tex]\[ \frac{x+1}{x(x+2)} \][/tex]
Step 3: Subtract the two fractions:
[tex]\[ \frac{x(x+5)}{x(x+2)} - \frac{x+1}{x(x+2)} = \frac{x(x+5) - (x+1)}{x(x+2)} \][/tex]
Step 4: Simplify the numerator:
[tex]\[
x(x+5) - (x+1) = x^2 + 5x - x - 1 = x^2 + 4x - 1
\][/tex]
Thus, the expression simplifies to:
[tex]\[ \frac{x^2 + 4x - 1}{x(x+2)} \][/tex]
We now match this result with the options given:
1. [tex]\(\frac{x^2+4 x-1}{x(x+2)}\)[/tex]
2. [tex]\(\frac{x^2+4 x+1}{x(x+2)}\)[/tex]
3. [tex]\(\frac{4}{-1\left(x^2+x-2\right)}\)[/tex]
4. [tex]\(\frac{x^2+6 x+1}{x(x+2)}\)[/tex]
The correct option is clearly:
[tex]\[ \frac{x^2+4 x-1}{x(x+2)} \][/tex]
Therefore, the difference is:
[tex]\[ \boxed{\frac{x^2+4 x-1}{x(x+2)}} \][/tex]
