Simplify ((x-2)/(x^2-4))÷((x+1)/(x^2-2x-3))

$(\frac{x - 2}{x^{2} - 4}) \div (\frac{x + 1}{x^{2} - 2 x - 3})$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x - 2}{x^{2} - 4} \cdot \frac{x^{2} - 2 x - 3}{x + 1}$

Step 2

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

Step 3

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Factor $x^{2} - 2 x - 3$ 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Write the factored form using these integers.

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

Step 5

Simplify terms.

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Divide $x - 3$ by $1$ .

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

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

$\frac{x - 3}{x + 2}$

Simplify ((x-2)/(x^2-4))÷((x+1)/(x^2-2x-3))

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