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

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

Multiply the numerator by the reciprocal of the denominator.

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

Simplify the denominator.

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

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

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 $b = 2$ .

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

Factor using the perfect square rule.

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

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

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

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

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

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

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

Simplify terms.

Combine.

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

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Move $3$ to the left of $x + 2$ .

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