To simplify the given rational expression, let's analyze the expression step by step. The expression provided is:

$\frac{2x^2 - x - 1}{x^2 - 9} \cdot \frac{x + 3}{2x + 1}$

Step 1: Factor each part of the expression.

1. Numerator of the first fraction: $2x^2 - x - 1$

   • This can be factored by splitting the middle term or by trial and error. We find that: $2x^2 - x - 1 = (2x + 1)(x - 1)$

2. Denominator of the first fraction: $x^2 - 9$

   • This is a difference of squares, which factors as: $x^2 - 9 = (x + 3)(x - 3)$

3. Numerator of the second fraction: $x + 3$

   • This is already in its simplest form.

4. Denominator of the second fraction: $2x + 1$

   • This is also in its simplest form.

Step 2: Substitute the factors into the expression.

Now, substitute these factors into the original expression:

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

Step 3: Simplify the expression.

Now, cancel out the common factors in the numerator and the denominator.

• The factor $2x + 1$ in the numerator of the first fraction and the denominator of the second fraction cancels out.
• The factor $x + 3$ in the denominator of the first fraction and the numerator of the second fraction cancels out.

This leaves us with:

$\frac{x - 1}{x - 3}$

Final Answer:

The simplified form of the given rational expression is:

$\frac{x - 1}{x - 3}$

Would you like more details on any part of the solution, or do you have any other questions?

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Here are some related questions:

1. How do you factor quadratic expressions like $2x^2 - x - 1$?
2. What are the common types of factoring techniques used in algebra?
3. How does the difference of squares factoring work?
4. How can rational expressions be simplified when multiple variables are involved?
5. What are the conditions under which a rational expression is undefined?

Tip: Always look for common factors in both the numerator and denominator before multiplying or dividing rational expressions. This can simplify the process significantly.