To simplify the given expression:

$\frac{6x}{(5x - 7)(x + 1)} - \frac{1}{5x - 7}$

Step 1: Find a common denominator.

The first term already has the denominator $(5x - 7)(x + 1)$, while the second term has only $5x - 7$ in the denominator. To subtract these fractions, we need to bring the second term to have the same denominator by multiplying its numerator and denominator by $x + 1$.

$\frac{1}{5x - 7} \cdot \frac{x + 1}{x + 1} = \frac{x + 1}{(5x - 7)(x + 1)}$

Now the expression becomes:

$\frac{6x}{(5x - 7)(x + 1)} - \frac{x + 1}{(5x - 7)(x + 1)}$

Step 2: Combine the fractions.

Now that both terms have the same denominator, we can combine the numerators:

$\frac{6x - (x + 1)}{(5x - 7)(x + 1)}$

Step 3: Simplify the numerator.

Simplify the numerator $6x - (x + 1)$:

$6x - x - 1 = 5x - 1$

Final expression:

$\frac{5x - 1}{(5x - 7)(x + 1)}$

This is the simplified form of the given expression.

---

Would you like further details or have any questions about this?

Here are 5 related questions to expand your understanding:

1. How do you simplify fractions with different denominators in general?
2. What are the steps to factor complex denominators?
3. How can you handle expressions with multiple variables in a similar situation?
4. What would happen if one of the terms in the expression were squared?
5. How do you find common denominators for rational expressions?

Tip: Always look for common factors or opportunities to factorize terms before attempting to simplify complex rational expressions!