Explicație pas cu pas:
a)
[tex]\frac{2 {x}^{2} + 4x}{ {x}^{2} + 4x + 4} - 1 = \frac{2x(x + 2)}{ {(x + 2)}^{2} } - 1 = \\ [/tex]
[tex]= \frac{2x}{x + 2} - 1 = \frac{2x - (x + 2)}{x + 2} = \frac{2x - x - 2}{x + 2} \\ [/tex]
[tex]= \bf \frac{x - 2}{x + 2} \\ [/tex]
b)
[tex]E(x) = \left( \frac{ {x}^{2} + 4}{2x} - 2 \right) : \left(\frac{2 {x}^{2} + 4x}{ {x}^{2} + 4x + 4} - 1 \right) \cdot \left( \frac{1}{x + 2} + \frac{1}{x - 2} \right) = \\ [/tex]
[tex]= \frac{ {x}^{2} + 4 - 4x}{2x} : \frac{x - 2}{x + 2} \cdot \frac{x - 2 + x + 2}{(x + 2)(x - 2)} \\ [/tex]
[tex]= \frac{ {(x - 2)}^{2}}{2x} \cdot \frac{x + 2}{x - 2} \cdot \frac{2x}{(x + 2)(x - 2)} \\ [/tex]
[tex]= \frac{ 2x \cdot (x + 2){(x - 2)}^{2}}{2x \cdot (x + 2){(x - 2)}^{2}} = \bf1 \\ [/tex]
[tex]\implies {\left[E(x) -2 \right]}^{100} = {(1 - 2)}^{100} = \\ = {( - 1)}^{100} = {1}^{100} = \red{\bf 1}[/tex]
