x/x^2+2x+1 + 3/4x^2-4
To simplify the expression (x/x^2+2x+1) + (3/4x^2-4), we can follow these steps:
Step 1: Factor the denominators of both fractions to identify any common factors.
The denominator of the first fraction, x^2+2x+1, can be factored as (x+1)(x+1) or (x+1)^2.
The denominator of the second fraction, 4x^2-4, can be factored as 4(x^2-1), and further as 4(x+1)(x-1).
Now we have the expression in factored form: x/(x+1)^2 + 3/(4(x+1)(x-1)).
Step 2: Find the least common denominator (LCD) of the two fractions.
The LCD is the smallest multiple of all the factors in the denominators of both fractions. In this case, the LCD is 4(x+1)(x-1) since it includes all the factors from both denominators.
Step 3: Rewrite each fraction with the common denominator.
To do this, we need to multiply the numerator and denominator of each fraction by any missing factors of the denominator.
The first fraction already has the common denominator. Its numerator is x, so it remains the same.
For the second fraction, we need to multiply the numerator and denominator by (x+1) to get the common denominator:
3/(4(x+1)(x-1)) * (x+1)/(x+1) = 3(x+1)/(4(x+1)(x-1)).
Now we have the expression with a common denominator: x/(x+1)^2 + 3(x+1)/(4(x+1)(x-1)).
Step 4: Combine the fractions over the common denominator.
Since we have fractions with the same denominator, we can add their numerators together and keep the common denominator:
[x + 3(x+1)] / [(x+1)^2 * 4(x+1)(x-1)] = [x + 3x + 3] / [(x+1)^2 * 4(x+1)(x-1)].
Simplifying the numerator, we get:
4x + 3 / [(x+1)^2 * 4(x+1)(x-1)].
Therefore, the simplified expression is (4x + 3) / [(x+1)^2 * 4(x+1)(x-1)].