We are given the following expression to multiply and simplify:

$\frac{(x^2 + x - 2)(xy^2)}{(x^2 + 4x + 4)(x^7y)}$

Step 1: Factor the quadratic expressions

First, let's factor the quadratic expressions in the numerator and the denominator.

• $x^2 + x - 2$ factors as $(x + 2)(x - 1)$
• $x^2 + 4x + 4$ factors as $(x + 2)(x + 2)$ or $(x + 2)^2$

Now, we rewrite the expression using the factored forms:

$\frac{(x + 2)(x - 1)(xy^2)}{(x + 2)^2(x^7y)}$

Step 2: Simplify by canceling common factors

We see that there is a common factor of $x + 2$ in both the numerator and denominator. Cancel one $x + 2$ from both:

$\frac{(x - 1)(xy^2)}{(x + 2)(x^7y)}$

Step 3: Combine like terms

Now, let's simplify the remaining expression by multiplying and canceling terms.

• Numerator: $(x - 1)(xy^2) = x(x - 1)y^2$
• Denominator: $(x + 2)(x^7y) = x^7(x + 2)y$

Now the expression becomes:

$\frac{x(x - 1)y^2}{x^7(x + 2)y}$

Step 4: Simplify the powers of $x$ and $y$

• $\frac{x}{x^7} = \frac{1}{x^6}$
• $\frac{y^2}{y} = y$

Thus, the simplified expression is:

$\frac{(x - 1)y}{x^6(x + 2)}$

Final Answer:

The simplified expression is:

$\frac{(x - 1)y}{x^6(x + 2)}$

Let me know if you need further clarification or have any questions!

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Here are some related questions:

1. How do you factor quadratic expressions?
2. What are the rules for simplifying powers of variables?
3. How do you cancel terms when simplifying fractions?
4. What is the method for multiplying polynomials?
5. How do you handle expressions with multiple variables?

Tip: Always look for common factors when simplifying expressions to make the process easier.