Let's simplify the given expression step by step:

The original expression is:

$\frac{(x + 5)(x - 1)}{3(x + 6)} \times \frac{2x - 1}{x + 5}$

Step 1: Cancel out common terms

Notice that $(x + 5)$ appears in both the numerator and denominator, so we can cancel these terms. This simplifies the expression to:

$\frac{(x - 1)}{3(x + 6)} \times (2x - 1)$

Step 2: Multiply the numerators and denominators

Now, multiply the remaining terms:

$\text{Numerator: } (x - 1)(2x - 1)$ $\text{Denominator: } 3(x + 6)$

Thus, the simplified expression becomes:

$\frac{(x - 1)(2x - 1)}{3(x + 6)}$

Step 3: Compare with the given options

Now, let's compare the simplified expression with the answer choices:

• Option (a): $\frac{(x - 1)(2x + 1)}{3(x + 6)}$ (Incorrect, because we have $2x - 1$, not $2x + 1$).
• Option (b): $\frac{(x - 1)(2x - 1)}{3}$ (Incorrect, because the denominator does not have $(x + 6)$).
• Option (c): $\frac{(x - 1)(2x - 1)}{3(x + 6)}$ (Correct, this matches the simplified expression).
• Option (d): $\frac{(x + 1)(2x - 1)}{3(x + 6)}$ (Incorrect, because we have $x - 1$, not $x + 1$).

Final Answer:

The correct answer is (c) $\frac{(x - 1)(2x - 1)}{3(x + 6)}$.

Would you like further clarification or details on any step? Here are a few follow-up questions that could help expand the topic:

1. How do you simplify rational expressions by factoring?
2. What are the rules for canceling terms in a fraction?
3. What happens when you multiply two fractions?
4. How can you identify common factors in algebraic expressions?
5. What is the process for simplifying complex rational expressions with multiple terms?

Tip: Always look for common factors between the numerator and denominator when simplifying rational expressions to make the process quicker and easier!