Certainly! Let's perform the operation and combine the given fractions into one.
First, let's analyze the given fractions:
[tex]\[
\frac{2}{x+5} \quad \text{and} \quad \frac{3x}{x^2 - 2x - 35}
\][/tex]
Our goal is to find a common denominator and combine these fractions.
### Step 1: Factorize the denominator in the second fraction
The denominator of the second fraction is [tex]\( x^2 - 2x - 35 \)[/tex]. We need to factorize this quadratic expression:
[tex]\[
x^2 - 2x - 35 = (x - 7)(x + 5)
\][/tex]
So, the second fraction can be written as:
[tex]\[
\frac{3x}{(x - 7)(x + 5)}
\][/tex]
### Step 2: Find a common denominator
The common denominator of the two fractions [tex]\(\frac{2}{x+5}\)[/tex] and [tex]\(\frac{3x}{(x-7)(x+5)}\)[/tex] is the least common multiple (LCM) of [tex]\(x + 5\)[/tex] and [tex]\((x - 7)(x + 5)\)[/tex]. Hence, the common denominator is:
[tex]\[
(x - 7)(x + 5)
\][/tex]
### Step 3: Rewrite each fraction with the common denominator
To rewrite each fraction with this common denominator, we need to adjust the numerators:
#### First fraction:
[tex]\[
\frac{2}{x + 5} = \frac{2 \cdot (x - 7)}{(x + 5) \cdot (x - 7)} = \frac{2(x - 7)}{(x - 7)(x + 5)}
\][/tex]
#### Second fraction:
This fraction already has the desired denominator:
[tex]\[
\frac{3x}{(x - 7)(x + 5)}
\][/tex]
### Step 4: Combine the fractions
Now that both fractions have the same denominator, we can combine them:
[tex]\[
\frac{2(x - 7)}{(x - 7)(x + 5)} + \frac{3x}{(x - 7)(x + 5)} = \frac{2(x - 7) + 3x}{(x - 7)(x + 5)}
\][/tex]
### Step 5: Simplify the numerator
Combine like terms in the numerator:
[tex]\[
2(x - 7) + 3x = 2x - 14 + 3x = 2x + 3x - 14 = 5x - 14
\][/tex]
So, the combined fraction becomes:
[tex]\[
\frac{5x - 14}{(x - 7)(x + 5)}
\][/tex]
### Step 6: Simplify the final expression (if possible)
Let's rewrite the denominator for clarity:
[tex]\[
(x - 7)(x + 5) = x^2 - 2x - 35
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{5x - 14}{x^2 - 2x - 35}
\][/tex]
Therefore, the combined fraction [tex]\(\frac{2}{x+5} + \frac{3x}{x^2 - 2x - 35}\)[/tex] simplifies to:
[tex]\[
\boxed{\frac{2x^2 - 25x + 98}{x^2 - 2x - 35}}
\][/tex]
This completes the operation.
