[tex]x^2-4=(x-2)(x+2)\\2-x=-(x-2)\\\\D:x+2\not=0 \wedge x-2\not=0\\D:x\not =-2 \wedge x\not=2\\\\\\\dfrac{x}{x+2}-\dfrac{1}{2-x}=\dfrac{4}{x^2-4}\\\\\dfrac{x}{x+2}+\dfrac{1}{x-2}=\dfrac{4}{(x-2)(x+2)}|\cdot (x-2)(x+2)\\\\x(x-2)+x+2=4\\x^2-2x+x-2=0\\x^2-x-2=0\\x^2+x-2x-2=0\\x(x+1)-2(x+1)=0\\(x-2)(x+1)=0\\x=2 \vee x=-1\\2\not\in D\\\\\boxed{x=-1}[/tex]
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[tex]x^2-4=(x-2)(x+2)\\2-x=-(x-2)\\\\D:x+2\not=0 \wedge x-2\not=0\\D:x\not =-2 \wedge x\not=2\\\\\\\dfrac{x}{x+2}-\dfrac{1}{2-x}=\dfrac{4}{x^2-4}\\\\\dfrac{x}{x+2}+\dfrac{1}{x-2}=\dfrac{4}{(x-2)(x+2)}|\cdot (x-2)(x+2)\\\\x(x-2)+x+2=4\\x^2-2x+x-2=0\\x^2-x-2=0\\x^2+x-2x-2=0\\x(x+1)-2(x+1)=0\\(x-2)(x+1)=0\\x=2 \vee x=-1\\2\not\in D\\\\\boxed{x=-1}[/tex]