Perform the indicated operations and simplify.
20y2 − 9y − 20
20y2 + 41y + 20
 ÷ 
20y2 − 41y + 20
25y2 − 40y + 16


Let's perform the indicated operations and simplify the given expression:

### The expression is:
\[
\frac{(20y^2 - 9y - 20)(20y^2 + 41y + 20)}{(20y^2 - 41y + 20)(25y^2 - 40y + 16)}
\]

### Step 1: Factor each quadratic expression.

#### 1. \(20y^2 - 9y - 20\)
We factor this quadratic:
\[
20y^2 - 9y - 20 = (4y + 5)(5y - 4)
\]

#### 2. \(20y^2 + 41y + 20\)
We factor this quadratic:
\[
20y^2 + 41y + 20 = (4y + 5)(5y + 4)
\]

#### 3. \(20y^2 - 41y + 20\)
We factor this quadratic:
\[
20y^2 - 41y + 20 = (4y - 5)(5y - 4)
\]

#### 4. \(25y^2 - 40y + 16\)
We factor this quadratic:
\[
25y^2 - 40y + 16 = (5y - 4)^2
\]

### Step 2: Rewrite the expression with factored terms.
\[
\frac{(4y + 5)(5y - 4)(4y + 5)(5y + 4)}{(4y - 5)(5y - 4)(5y - 4)^2}
\]

### Step 3: Simplify the expression.
Cancel out the common factors from the numerator and the denominator:
- \(5y - 4\) appears once in the numerator and twice in the denominator, so cancel one instance.
- No other common terms cancel directly.

We are left with:
\[
\frac{(4y + 5)(4y + 5)(5y + 4)}{(4y - 5)(5y - 4)}
\]

This simplifies to:
\[
\frac{(4y + 5)^2(5y + 4)}{(4y - 5)(5y - 4)}
\]

### Final Answer:
\[
\frac{(4y + 5)^2(5y + 4)}{(4y - 5)(5y - 4)}
\]

Let me know if you would like any details or have further questions!

Here are 5 related questions for practice:
1. Factor \(x^2 + 6x + 9\).
2. Simplify \(\frac{(x - 2)(x + 4)}{(x - 2)(x + 5)}\).
3. Factor \(2x^2 + 5x - 3\).
4. Solve \(\frac{(x + 1)(x - 3)}{(x + 1)(x + 5)} = 0\).
5. Simplify \(\frac{(x + 7)(x - 2)}{(x + 7)(x + 3)}\).

**Tip:** Always check for common factors before multiplying or simplifying expressions.

Perform the indicated operations and simplify. 
20y^2 − 9y − 20 
20y^2 + 41y + 20 ÷ 20y^2 − 41y + 20 
25y^2 − 40y + 16

Learn how to perform operations and simplify a complex rational expression involving quadratics: (20y^2 - 9y - 20)(20y^2 + 41y + 20) ÷ (20y^2 - 41y + 20)(25y^2 - 40y + 16). This step-by-step solution covers factoring quadratics and simplifying rational expressions. Suitable for high school students in Grades 9-11.

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