To solve the given problem, we need to perform the indicated operation (division) between two rational expressions and then simplify the result.

The problem is:

$\frac{x^4 - 5x^3}{4x^2 - x - 3} \div \frac{x^3 - 25x}{16x^2 - 9}$

Step 1: Convert Division to Multiplication

To divide two fractions, we multiply the first fraction by the reciprocal of the second:

$\frac{x^4 - 5x^3}{4x^2 - x - 3} \times \frac{16x^2 - 9}{x^3 - 25x}$

Step 2: Factor All Polynomials

Let's factor each polynomial:

1. Numerator of the first fraction: $x^4 - 5x^3$:

$x^4 - 5x^3 = x^3(x - 5)$

2. Denominator of the first fraction: $4x^2 - x - 3$:

We use factoring by grouping or the quadratic formula. Factoring gives:

$4x^2 - x - 3 = (4x + 3)(x - 1)$

3. Numerator of the second fraction: $16x^2 - 9$:

This is a difference of squares:

$16x^2 - 9 = (4x - 3)(4x + 3)$

4. Denominator of the second fraction: $x^3 - 25x$:

Factor out the greatest common factor, $x$:

$x^3 - 25x = x(x^2 - 25) = x(x - 5)(x + 5)$

Step 3: Substitute the Factored Forms

Now substitute the factored forms back into the expression:

$\frac{x^3(x - 5)}{(4x + 3)(x - 1)} \times \frac{(4x - 3)(4x + 3)}{x(x - 5)(x + 5)}$

Step 4: Simplify by Canceling Common Factors

Cancel out common factors in the numerator and the denominator:

• $x^3$ and $x$ simplify to $x^2$
• $x - 5$ in the numerator and denominator cancel out
• $4x + 3$ in the numerator and denominator cancel out

The simplified expression becomes:

$\frac{x^2(4x - 3)}{(x - 1)(x + 5)}$

Final Simplified Form

The final simplified form of the expression is:

$\frac{x^2(4x - 3)}{(x - 1)(x + 5)}$

Would you like further clarification or details on any part of this solution?

Here are some follow-up questions for deeper understanding:

1. How do you handle division of rational expressions in general?
2. What are the rules for factoring polynomials such as differences of squares or trinomials?
3. Can you explain how to identify and cancel common factors in a rational expression?
4. How do you determine the domain of the resulting simplified expression?
5. Why is it important to always factor completely before simplifying?

Tip:

Always check for any restrictions on the variables by identifying values that make the original denominators zero.