We are tasked with dividing the expression

$\frac{2x^3 + 2x^2 - 4x}{x^3 + 4x^2 - 5x}.$

Step 1: Factor both numerator and denominator

Factor the numerator: $2x^3 + 2x^2 - 4x$

1. Take out the common factor $2x$: $2x(x^2 + x - 2).$
2. Factor $x^2 + x - 2$ (using the quadratic formula or factoring): $x^2 + x - 2 = (x + 2)(x - 1).$ Thus, the numerator becomes: $2x(x + 2)(x - 1).$

Factor the denominator: $x^3 + 4x^2 - 5x$

1. Take out the common factor $x$: $x(x^2 + 4x - 5).$
2. Factor $x^2 + 4x - 5$ (using the quadratic formula or factoring): $x^2 + 4x - 5 = (x + 5)(x - 1).$ Thus, the denominator becomes: $x(x + 5)(x - 1).$

Step 2: Simplify the fraction

The fraction becomes: $\frac{2x(x + 2)(x - 1)}{x(x + 5)(x - 1)}.$

Cancel out the common factors:

1. $x$ (as long as $x \neq 0$),
2. $(x - 1)$ (as long as $x \neq 1$).

The simplified expression is: $\frac{2(x + 2)}{x + 5}, \quad \text{with the restrictions that } x \neq 0, x \neq 1, \text{ and } x \neq -5.$

Final Answer:

$\frac{2(x + 2)}{x + 5}, \quad \text{where } x \neq 0, x \neq 1, \text{ and } x \neq -5.$

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Would you like me to verify the factorization or expand further?

Related Questions:

1. How do you factorize higher-degree polynomials?
2. What are the restrictions when simplifying rational expressions?
3. How do you handle cases where the numerator or denominator cannot be factored?
4. Can you graph this simplified function to visualize its restrictions?
5. How do vertical asymptotes relate to the restrictions on $x$ in rational functions?

Tip:

Always check for common factors before attempting long division in rational expressions. It saves time and simplifies calculations!