To simplify the expression:

$\left( \frac{y^2 - 2y}{y^2 + y - 2} \right) \times \left( \frac{y - 1}{y^2 + 4y + 4} \right) \div \left( \frac{y^2 + 5y - 14}{y^4} \right)$

We can break this down step-by-step.

Step 1: Factor all the quadratic expressions

Let's factor the quadratic expressions where possible:

1. $y^2 - 2y = y(y - 2)$
2. $y^2 + y - 2$ can be factored as $(y - 1)(y + 2)$
3. $y^2 + 4y + 4$ is a perfect square, so it factors as $(y + 2)^2$
4. $y^2 + 5y - 14$ factors as $(y - 2)(y + 7)$

Now, substitute the factored forms back into the expression:

$\left( \frac{y(y - 2)}{(y - 1)(y + 2)} \right) \times \left( \frac{y - 1}{(y + 2)^2} \right) \div \left( \frac{(y - 2)(y + 7)}{y^4} \right)$

Step 2: Simplify the multiplication

When multiplying fractions, we multiply the numerators together and the denominators together:

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

Notice that $(y - 1)$ cancels out:

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

This simplifies to:

$\frac{y(y - 2)}{(y + 2)^3}$

Step 3: Perform the division

Now we need to divide by $\frac{(y - 2)(y + 7)}{y^4}$. Dividing by a fraction is the same as multiplying by its reciprocal, so we multiply by $\frac{y^4}{(y - 2)(y + 7)}$:

$\frac{y(y - 2)}{(y + 2)^3} \times \frac{y^4}{(y - 2)(y + 7)}$

Cancel the common $(y - 2)$ terms:

$\frac{y \cdot y^4}{(y + 2)^3(y + 7)}$

Step 4: Final simplified expression

Now, multiply the numerators and combine the terms:

$\frac{y^5}{(y + 2)^3(y + 7)}$

This is the simplified expression.

Final Answer:

$\frac{y^5}{(y + 2)^3(y + 7)}$

Would you like more details on any part of this solution?

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Here are some related questions for further practice:

1. Factor the expression $y^2 + 6y + 9$.
2. Simplify $\frac{(x^2 - 4)}{(x + 2)} \times \frac{(x + 2)}{(x^2 - 1)}$.
3. Solve $\frac{y^2 + y - 6}{y^2 - 1}$.
4. Factor and simplify $\frac{y^2 + 7y + 12}{y^2 - 9}$.
5. Simplify the expression $\frac{(x^2 - 9)}{(x^2 - 1)} \div \frac{(x + 3)}{(x - 1)}$.

Tip: Always check for common factors to cancel before multiplying or dividing algebraic fractions.