Simplify:

x+2/x^2+x-2/2x

To simplify the given expression:

(x + 2) / (x^2 + x) - (2x - 2) / (2x)

Step 1: Factor the denominators:

For the first fraction, (x^2 + x), there are no common factors, so we cannot simplify it further.

For the second fraction, (2x), we can factor out 2x:

(2x - 2) = 2(x - 1)

Step 2: Combine the two fractions with the same denominator:

(x + 2) / (x^2 + x) - (2(x - 1)) / (2x)

Step 3: Find the least common denominator (LCD):

Since the two denominators do not have any common factors, the LCD is simply the product of the two denominators:

LCD = (x^2 + x) * (2x)

Step 4: Adjust the numerators to have the same denominator as the LCD:

For the first fraction, multiply the numerator and denominator by 2x:

[(x + 2) * 2x] / [(x^2 + x) * 2x] - [(2(x - 1)) * (x^2 + x)] / [(2x) * (x^2 + x)]

Simplifying the numerators:

2x^2 + 4x - [2(x^3 - x^2 + x^2 - x)] / [(x^2 + x) * 2x]

Step 5: Combine the numerators:

2x^2 + 4x - (2x^3 - 2x^2 + 2x - 2x) / [(x^2 + x) * 2x]

Simplifying further:

2x^2 + 4x - 2x^3 + 2x^2 - 2(x - 1) / [(x^2 + x) * 2x]

Step 6: Combine the like terms:

-2x^3 + 4x^2 + 4x - 2(x - 1) / [(x^2 + x) * 2x]

Step 7: Simplify the expression further, if possible:

Since there are no common factors that can be canceled out between the numerator and the denominator, the expression cannot be simplified any further.