Let's break down the given expression:

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

Step 1: Rewriting Division as Multiplication

We rewrite the division of two fractions as the multiplication of the first fraction by the reciprocal of the second fraction:

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

Step 2: Factoring Both Numerators and Denominators

We now factor all the terms.

1. Factoring $x^4 - 5x^3$: $x^4 - 5x^3 = x^3(x - 5)$

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

3. Factoring $4x^2 - x - 3$: We use the quadratic factorization method to factor this: $4x^2 - x - 3 = (4x + 3)(x - 1)$

4. Factoring $16x^2 - 9$: This is a difference of squares: $16x^2 - 9 = (4x - 3)(4x + 3)$

Step 3: Substituting the Factored Terms

Now substitute the factored forms 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: Canceling Common Factors

Now we cancel out the common factors from the numerator and denominator:

• $x - 5$ cancels out.
• $4x + 3$ cancels out.
• $x$ cancels out.

After canceling, we are left with:

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

Final Answer:

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

Would you like further clarification or details on any of the steps?

Here are some related questions to expand on the concept:

1. How do you factor a quadratic equation in general?
2. What is the difference of squares, and how is it used in factoring?
3. Can you explain how division of fractions works with algebraic expressions?
4. How can you simplify a rational expression?
5. What are some strategies for canceling terms in algebraic fractions?

Tip: Always check for common factors between numerators and denominators before multiplying or dividing rational expressions. This can simplify your work significantly.