91Ó°ÊÓ

Start with factoring the polynomials in the numerator and denominator of both expressions. \[ \frac{x^{2}-4}{x^{2}-4 x+4} \cdot \frac{2 x-4}{x+2} = \frac{(x-2)(x+2)}{(x-2)^2} \cdot \frac{2(x-2)}{(x+2)} \]

Next, cancel out the common terms in the numerator and the denominator of the resulted expressions. This simplification step is based on the principle that \(\frac{A}{A} = 1\) where \(A ≠ 0\). Also note that, when cancelling common factors, we must also specify the conditions related to the undefined operation \(division by zero\), which originate from the original denominator. Hence, from the expression, \(2(x-2), (x+2), (x-2)^2\) in the denominator haven’t to be equal to zero. We have the following conditions \(x ≠ 2, x ≠ -2\), however in the original equation we have only restriction \(x ≠ 2\), because \(x ≠ -2\) is already cancelled out. Thus, we have \[ \frac{(x-2)(x+2)}{(x-2)^2} \cdot \frac{2(x-2)}{(x+2)} = 2 \], provided that \(x ≠ 2 \)

We write the final simplified result of the given rational expressions. \(2\), \(x ≠ 2\) is our solution