Simplify (x-2)/(2x-3)*(4x-6)/(x^2-4)

$\frac{x - 2}{2 x - 3} \cdot \frac{4 x - 6}{x^{2} - 4}$

Step 1

Factor $2$ out of $4 x - 6$ .

Factor $2$ out of $4 x$ .

$\frac{x - 2}{2 x - 3} \cdot \frac{2 (2 x) - 6}{x^{2} - 4}$

Factor $2$ out of $- 6$ .

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

Factor $2$ out of $2 (2 x) + 2 (- 3)$ .

$\frac{x - 2}{2 x - 3} \cdot \frac{2 (2 x - 3)}{x^{2} - 4}$

Step 2

Simplify the denominator.

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

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

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 $b = 2$ .

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

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x - 2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2 x - 3$ .

Factor $2 x - 3$ out of $2 (2 x - 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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