We are given the following expression to multiply:

$\frac{4x^2 - 9x + 5}{x^2 - 1} \times \frac{x^2 + x}{4x^2 - x - 5}$

Step 1: Factor each part of the expression

Let's factor each of the numerators and denominators where possible.

Factor $4x^2 - 9x + 5$

We need to factor this quadratic trinomial. We try factoring by grouping.

$4x^2 - 9x + 5 = (4x - 5)(x - 1)$

Factor $x^2 - 1$

This is a difference of squares:

$x^2 - 1 = (x - 1)(x + 1)$

Factor $4x^2 - x - 5$

We try factoring this quadratic trinomial as well:

$4x^2 - x - 5 = (4x + 5)(x - 1)$

Step 2: Rewrite the expression with factored terms

Now that we've factored everything, rewrite the original expression:

$\frac{(4x - 5)(x - 1)}{(x - 1)(x + 1)} \times \frac{x(x + 1)}{(4x + 5)(x - 1)}$

Step 3: Cancel out common factors

We can cancel out the common terms from the numerator and denominator:

• $(x - 1)$ cancels out from both the numerator and the denominator.
• $(x + 1)$ cancels out from both the numerator and the denominator.
• $4x + 5$ cancels out from both the numerator and the denominator.

After canceling, we're left with:

$\frac{x}{1} = x$

Final Answer:

The simplified product is:

$x$

Let me know if you want further details or if you have any questions.

Here are 5 related questions:

1. How do you factor quadratic trinomials?
2. What is the difference of squares rule in factoring?
3. How do you simplify rational expressions?
4. What are the common factorization techniques?
5. How do you multiply and divide rational expressions?

Tip: Always look for common factors before multiplying or dividing rational expressions to simplify the process.