Perform the indicated operations and simplify the following expression.
StartFraction y squared minus 2 y Over y squared plus y minus 2 EndFraction times StartFraction y minus 1 Over y squared plus 4 y plus 4 EndFraction divided by StartFraction y squared plus 5 y minus 14 Over y Superscript 4 EndFraction
y2−2y
y2+y−2•
y−1
y2+4y+4÷
y2+5y−14
y4
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Part 1
StartFraction y squared minus 2 y Over y squared plus y minus 2 EndFraction times StartFraction y minus 1 Over y squared plus 4 y plus 4 EndFraction divided by StartFraction y squared plus 5 y minus 14 Over y Superscript 4 EndFraction
y2−2y
y2+y−2•
y−1
y2+4y+4÷
y2+5y−14
y4equals=enter your response here
​(Simplify your answer. Type your answer in factored​ form.)

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.

Perform the indicated operations and simplify the following expression: StartFraction y squared minus 2 y Over y squared plus y minus 2 EndFraction times StartFraction y minus 1 Over y squared plus 4 y plus 4 EndFraction divided by StartFraction y squared plus 5 y minus 14 Over y Superscript 4 EndFraction.

Learn how to simplify a complex algebraic expression involving factoring and rational expressions. This detailed guide walks through each step, from factoring quadratics to simplifying the final expression. Suitable for high school students in Grades 9-12.

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