Simplify ((9y^2)/(y^2+2y+1))÷((36y)/(y^2-1))

$\frac{9 y^{2}}{y^{2} + 2 y + 1} \div \frac{36 y}{y^{2} - 1}$

To divide by a fraction, multiply by its reciprocal.

$\frac{9 y^{2}}{y^{2} + 2 y + 1} \cdot \frac{y^{2} - 1}{36 y}$

Simplify terms.

Cancel the common factor of $9 y$ .

Factor $9 y$ out of $9 y^{2}$ .

$\frac{9 y (y)}{y^{2} + 2 y + 1} \cdot \frac{y^{2} - 1}{36 y}$

Factor $9 y$ out of $36 y$ .

$\frac{9 y (y)}{y^{2} + 2 y + 1} \cdot \frac{y^{2} - 1}{9 y (4)}$

Cancel the common factor.

$\frac{9 y \cdot y}{y^{2} + 2 y + 1} \cdot \frac{y^{2} - 1}{9 y \cdot 4}$

Rewrite the expression.

$\frac{y}{y^{2} + 2 y + 1} \cdot \frac{y^{2} - 1}{4}$

Multiply $\frac{y}{y^{2} + 2 y + 1}$ by $\frac{y^{2} - 1}{4}$ .

$\frac{y (y^{2} - 1)}{(y^{2} + 2 y + 1) \cdot 4}$

Simplify the numerator.

Rewrite $1$ as $1^{2}$ .

$\frac{y (y^{2} - 1^{2})}{(y^{2} + 2 y + 1) \cdot 4}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$\frac{y (y + 1) (y - 1)}{(y^{2} + 2 y + 1) \cdot 4}$

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$\frac{y (y + 1) (y - 1)}{(y^{2} + 2 y + 1^{2}) \cdot 4}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 y = 2 \cdot y \cdot 1$

Rewrite the polynomial.

$\frac{y (y + 1) (y - 1)}{(y^{2} + 2 \cdot y \cdot 1 + 1^{2}) \cdot 4}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 1$ .

$\frac{y (y + 1) (y - 1)}{{(y + 1)}^{2} \cdot 4}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $y + 1$ and ${(y + 1)}^{2}$ .

Factor $y + 1$ out of $y (y + 1) (y - 1)$ .

$\frac{(y + 1) (y (y - 1))}{{(y + 1)}^{2} \cdot 4}$

Cancel the common factors.

Factor $y + 1$ out of ${(y + 1)}^{2} \cdot 4$ .

$\frac{(y + 1) (y (y - 1))}{(y + 1) ((y + 1) \cdot 4)}$

Cancel the common factor.

$\frac{(y + 1) (y (y - 1))}{(y + 1) ((y + 1) \cdot 4)}$

Rewrite the expression.

$\frac{y (y - 1)}{(y + 1) \cdot 4}$

Move $4$ to the left of $y + 1$ .

$\frac{y (y - 1)}{4 (y + 1)}$