[tex] E(x) = (x^2 -1) : \left( x + \dfrac{x^2 -4}{2x-4} \cdot \dfrac{2x+4}{x^2+4x+4} \right) \\ E(x)= (x^2 -1) : \left( x+ \dfrac{(x-2)(x+2)}{2(x-2)} \cdot \dfrac{2(x+2)}{(x+2)(x+2)} \right) \\ E(x)= (x^2 -1) : \left( x+\dfrac{x+2}{2} \cdot \dfrac{2}{(x+2)} \right) \\ E(x) = (x^2-1): (x+1) \\ E(x)= \dfrac{x^2-1}{x+1} =\dfrac{(x+1)(x-1)}{x+1} =x-1 \\ \implies \tt E(x)= x-1, \ \forall x \in \mathbb{R} - \{-2,-1,2\} [/tex]
Păi pentru orice [tex] x \in \mathbb{N} ^{*} -\{2\} [/tex] expresia E(x)=x-1 este număr natural.
[tex] \implies E(x) \in \mathbb{N} \forall x\in \mathbb{N} ^{*} -\{2\} [/tex]
