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

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

Simplify the numerator.

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

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

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 $b = 1$ .

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

Factor $x^{2} + 3 x + 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

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

Factor using the perfect square rule.

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

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

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.

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

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

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

Simplify terms.

Cancel the common factor of $x - 1$ .

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

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

Factor $x - 1$ out of ${(x - 1)}^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 2$ .

Factor $x + 2$ out of $(x + 1) (x + 2)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 1}{x + 1} \cdot \frac{1}{x - 1}$

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

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

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{x - 1}$

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

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