5/x^2-4+2/x^2-4x+4-3/x^2-x-6

What are we doing ?

did you mean
5/(x^2 - 4) + 2/(x^2 - 4x - 3) - 3/(x^2 - x - 6)

Yes,I need to simplify it and find least common denamotator.

denaminator* (LCD)

first denominator = (x-2)(x+2)

second denominator = (x-2)(x-2) note Reiny typo
third denominator = (x-3)(x+2)
so
Least common denominator = (x-2)(x-2)(x+2)(x-3)

Please help me solve it

To simplify the expression 5/x^2-4 + 2/x^2-4x+4 - 3/x^2-x-6, we can follow these steps:

Step 1: Factorize each of the denominators:

The first denominator is x^2 - 4, which can be factored as (x + 2)(x - 2).
The second denominator is x^2 - 4x + 4, which can be factored as (x - 2)(x - 2) or (x - 2)^2.
The third denominator is x^2 - x - 6, which can be factored as (x - 3)(x + 2).

Now, let's write the expression with the factored denominators:

5/[(x + 2)(x - 2)] + 2/[(x - 2)(x - 2)] - 3/[(x - 3)(x + 2)]

Step 2: Find the least common denominator (LCD) for the expression. To do this, we need to multiply all the denominators together, including their common factors.

The common factors are (x + 2), (x - 2), and (x - 3).

So, the LCD = (x + 2)(x - 2)(x - 3)

Step 3: Rewrite the original expression with the LCD as the common denominator:

[(5(x - 3)) + (2(x + 2)) - (3(x - 2)(x - 3))]/[(x + 2)(x - 2)(x - 3)]

Simplifying further, we have:

(5x - 15 + 2x + 4 - 3x^2 + 15x - 3x + 6 )/[(x + 2)(x - 2)(x - 3)]

Combine like terms in the numerator:

(-3x^2 + 22x - 5)/[(x + 2)(x - 2)(x - 3)]

Therefore, the simplified form of the expression 5/x^2-4 + 2/x^2-4x+4 - 3/x^2-x-6 is (-3x^2 + 22x - 5)/[(x + 2)(x - 2)(x - 3)].