To simplify the expression [tex]\(\frac{6}{x^2 - 4} + \frac{x}{x + 2}\)[/tex], let's go through the steps in detail:
### Step 1: Factorization
First, observe that [tex]\(x^2 - 4\)[/tex] can be factored:
[tex]\[x^2 - 4 = (x - 2)(x + 2).\][/tex]
So, the given expression becomes:
[tex]\[\frac{6}{(x-2)(x+2)} + \frac{x}{x+2}.\][/tex]
### Step 2: Expressing with Common Denominator
In order to add these fractions, we need a common denominator. The common denominator will be [tex]\((x - 2)(x + 2)\)[/tex]:
The first fraction is already in the common denominator:
[tex]\[\frac{6}{(x-2)(x+2)}.\][/tex]
For the second fraction, [tex]\(\frac{x}{x+2}\)[/tex], we need to multiply both the numerator and the denominator by [tex]\((x-2)\)[/tex] to get the same denominator:
[tex]\[\frac{x \cdot (x - 2)}{(x + 2)(x - 2)} = \frac{x(x - 2)}{(x - 2)(x + 2)} = \frac{x^2 - 2x}{(x - 2)(x + 2)}.\][/tex]
### Step 3: Adding the Fractions
Now that both fractions have the same denominator, we can add their numerators:
[tex]\[\frac{6}{(x-2)(x+2)} + \frac{x^2 - 2x}{(x-2)(x+2)} = \frac{6 + (x^2 - 2x)}{(x-2)(x+2)}.\][/tex]
### Step 4: Simplifying the Numerator
Combine the terms in the numerator:
[tex]\[ 6 + x^2 - 2x = x^2 - 2x + 6.\][/tex]
So, the combined fraction is:
[tex]\[\frac{x^2 - 2x + 6}{(x - 2)(x + 2)}.\][/tex]
### Step 5: Result
Thus, the simplified form of the expression [tex]\(\frac{6}{x^2 - 4} + \frac{x}{x + 2}\)[/tex] is:
[tex]\[\frac{x^2 - 2x + 6}{x^2 - 4}.\][/tex]
To state it in the given format, your final simplified result would be:
[tex]\[
\frac{x^2 - 2x + 6}{x^2 - 4}.
\][/tex]
Here, the numerator of the simplified form is [tex]\(x^2 - 2x + 6\)[/tex] and the denominator remains [tex]\(x^2 - 4\)[/tex].
