Perform the indicated operation.
StartFraction x Over x squared minus 25 EndFraction
plusStartFraction 5 Over x plus 5 EndFraction
minusStartFraction 6 Over x EndFraction

To solve the expression 

\[
\frac{x}{x^2 - 25} + \frac{5}{x + 5} - \frac{6}{x},
\]

we first need to simplify the fractions by finding a common denominator. Here's the step-by-step process:

### Step 1: Factor the denominators
The first fraction has the denominator \(x^2 - 25\), which is a difference of squares. This can be factored as:

\[
x^2 - 25 = (x - 5)(x + 5).
\]

So the expression becomes:

\[
\frac{x}{(x - 5)(x + 5)} + \frac{5}{x + 5} - \frac{6}{x}.
\]

### Step 2: Determine the common denominator
The common denominator for all the fractions will be the product of the distinct factors from the denominators:

\[
\text{Common Denominator} = x(x + 5)(x - 5).
\]

### Step 3: Rewrite each fraction with the common denominator

1. **First fraction:**
   \[
   \frac{x}{(x - 5)(x + 5)} = \frac{x \cdot x}{x(x + 5)(x - 5)} = \frac{x^2}{x(x + 5)(x - 5)}.
   \]

2. **Second fraction:**
   \[
   \frac{5}{x + 5} = \frac{5 \cdot x(x - 5)}{x(x + 5)(x - 5)} = \frac{5x(x - 5)}{x(x + 5)(x - 5)}.
   \]

3. **Third fraction:**
   \[
   \frac{6}{x} = \frac{6 \cdot (x + 5)(x - 5)}{x(x + 5)(x - 5)} = \frac{6(x + 5)(x - 5)}{x(x + 5)(x - 5)}.
   \]

### Step 4: Combine the fractions
Now that all fractions have the same denominator, we can combine them:

\[
\frac{x^2}{x(x + 5)(x - 5)} + \frac{5x(x - 5)}{x(x + 5)(x - 5)} - \frac{6(x + 5)(x - 5)}{x(x + 5)(x - 5)}.
\]

Combine the numerators:

\[
\frac{x^2 + 5x(x - 5) - 6(x + 5)(x - 5)}{x(x + 5)(x - 5)}.
\]

### Step 5: Simplify the numerator
Expand and simplify the numerator:

1. Expand \(5x(x - 5)\):
   \[
   5x(x - 5) = 5x^2 - 25x.
   \]

2. Expand \(-6(x + 5)(x - 5)\):
   \[
   -6[(x + 5)(x - 5)] = -6(x^2 - 25) = -6x^2 + 150.
   \]

3. Combine all terms:
   \[
   x^2 + 5x^2 - 25x - 6x^2 + 150 = (x^2 + 5x^2 - 6x^2) - 25x + 150 = x^2 - 25x + 150.
   \]

### Final Answer:
Thus, the simplified form of the expression is:

\[
\frac{x^2 - 25x + 150}{x(x + 5)(x - 5)}.
\]

Would you like any further details or have any questions?

Here are some related questions to expand your understanding:
1. How do you factor a quadratic expression like \(x^2 - 25\)?
2. What is the significance of finding a common denominator in adding fractions?
3. Can the numerator \(x^2 - 25x + 150\) be factored further?
4. How does the difference of squares factoring help in simplifying rational expressions?
5. How do you solve an equation that involves a rational expression like the one we simplified?

**Tip:** Always factor expressions where possible before attempting to find a common denominator; this simplifies the process of combining fractions.

Learn how to simplify the rational expression (x / (x^2 - 25)) + (5 / (x + 5)) - (6 / x) with a detailed step-by-step solution. This problem covers key concepts in rational expressions, including finding a common denominator and factoring. Suitable for high school students.

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