Answer
$\displaystyle \frac{3}{x-2}$
Work Step by Step
$...=\displaystyle \frac{x-2}{4x}\div\frac{x^{2}-4x+4}{12x}=\frac{x-2}{4x}\cdot\frac{12x}{x^{2}-4x+4}$
Factoring,
$12=4\cdot 3$
$x^{2}-4x+4$ is a trinomial. We either recognize it as a perfect square of (x-2),
or we factor it by searching for factors of 4, whose sum is -4. These are $-2$ and $-2$.
$x^{2}-4x+4=(x-2)(x-2)$
... = $\displaystyle \frac{(x-2)\cdot 4\cdot 3x}{4\cdot x\cdot(x-2)(x-2)}$
... cancel the following from both sides of the fraction line:
$4,\ x$, $(x-2)$
... = $\displaystyle \frac{3}{x-2}$
