Simplify ((x^2)/(x^2+2x+1))÷((3x)/(x^2-1))

$\frac{x^{2}}{x^{2} + 2 x + 1} \div \frac{3 x}{x^{2} - 1}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2}}{x^{2} + 2 x + 1} \cdot \frac{x^{2} - 1}{3 x}$

Simplify terms.

Cancel the common factor of $x$ .

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x}{x^{2} + 2 x + 1} \cdot \frac{x^{2} - 1}{3 x}$

Factor $x$ out of $3 x$ .

$\frac{x \cdot x}{x^{2} + 2 x + 1} \cdot \frac{x^{2} - 1}{x \cdot 3}$

Cancel the common factor.

$\frac{x \cdot x}{x^{2} + 2 x + 1} \cdot \frac{x^{2} - 1}{x \cdot 3}$

Rewrite the expression.

$\frac{x}{x^{2} + 2 x + 1} \cdot \frac{x^{2} - 1}{3}$

Multiply $\frac{x}{x^{2} + 2 x + 1}$ by $\frac{x^{2} - 1}{3}$ .

$\frac{x (x^{2} - 1)}{(x^{2} + 2 x + 1) \cdot 3}$

Simplify the numerator.

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

$\frac{x (x^{2} - 1^{2})}{(x^{2} + 2 x + 1) \cdot 3}$

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

$\frac{x (x + 1) (x - 1)}{(x^{2} + 2 x + 1) \cdot 3}$

Factor using the perfect square rule.

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

$\frac{x (x + 1) (x - 1)}{(x^{2} + 2 x + 1^{2}) \cdot 3}$

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

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

$\frac{x (x + 1) (x - 1)}{(x^{2} + 2 \cdot x \cdot 1 + 1^{2}) \cdot 3}$

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

$\frac{x (x + 1) (x - 1)}{{(x + 1)}^{2} \cdot 3}$

Reduce the expression by cancelling the common factors.

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

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

$\frac{(x + 1) (x (x - 1))}{{(x + 1)}^{2} \cdot 3}$

Cancel the common factors.

Factor $x + 1$ out of ${(x + 1)}^{2} \cdot 3$ .

$\frac{(x + 1) (x (x - 1))}{(x + 1) ((x + 1) \cdot 3)}$

Cancel the common factor.

$\frac{(x + 1) (x (x - 1))}{(x + 1) ((x + 1) \cdot 3)}$

Rewrite the expression.

$\frac{x (x - 1)}{(x + 1) \cdot 3}$

Move $3$ to the left of $x + 1$ .

$\frac{x (x - 1)}{3 (x + 1)}$