The given expression is:

\[
\frac{5x}{5x^2 - 125} + \frac{5x - 5}{x^2 - 6x + 5}
\]

Let's simplify each term step-by-step.

### Step 1: Simplify \(\frac{5x}{5x^2 - 125}\)
Factor the denominator:

\[
5x^2 - 125 = 5(x^2 - 25) = 5(x - 5)(x + 5)
\]

So the expression becomes:

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

### Step 2: Simplify \(\frac{5x - 5}{x^2 - 6x + 5}\)
Factor the numerator and denominator:

\[
5x - 5 = 5(x - 1)
\]

\[
x^2 - 6x + 5 = (x - 5)(x - 1)
\]

So the expression becomes:

\[
\frac{5(x - 1)}{(x - 5)(x - 1)} = \frac{5}{x - 5}
\]

### Step 3: Combine the expressions
Now, we have:

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

To add these fractions, we need a common denominator. The common denominator will be \((x - 5)(x + 5)\). Rewrite the second fraction to have this common denominator:

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

Now, the expression becomes:

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

### Step 4: Add the fractions
Combine the numerators:

\[
\frac{x + 5(x + 5)}{(x - 5)(x + 5)} = \frac{x + 5x + 25}{(x - 5)(x + 5)} = \frac{6x + 25}{(x - 5)(x + 5)}
\]

So the simplified expression is:

\[
\frac{6x + 25}{(x - 5)(x + 5)}
\]

### Final Answer:
The simplified form of the given expression is:

\[
\frac{6x + 25}{(x - 5)(x + 5)}
\]

---

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you factor quadratic expressions like \(x^2 - 25\)?
2. What is the process of finding a common denominator for algebraic fractions?
3. Can you explain how to simplify complex rational expressions?
4. What are the differences between adding fractions with numerical versus algebraic terms?
5. How would this expression change if different terms were substituted into the equation?

**Tip:** Always factor expressions as much as possible before attempting to add or subtract them. It makes finding common denominators much easier!

Learn how to simplify the rational expression 5x/(5x^2 - 125) + (5x - 5)/(x^2 - 6x + 5) with step-by-step instructions. This problem involves factoring quadratics, simplifying fractions, and combining algebraic terms. Ideal for high school students in Grades 9-12.

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