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a/ có \(P=\left(\frac{2+x}{2-x}-\frac{4x^2}{4-x^2}-\frac{2-x}{2+x}\right).\frac{2x-x^2}{x-3}=\left(\frac{\left(2+x\right)^2-4x^2\left(4-x^2\right)-\left(2-x\right)^2}{4-x^2}\right).\frac{2x-x^2}{x-3}=\frac{4x^2+8x}{4-x^2}.\frac{2x-x^2}{x-3}=\frac{4x}{2-x}.\frac{x\left(2-x\right)}{x-3}=\frac{4x^2}{x-3}\)
a) ĐKXĐ: \(x\ne\pm2\)
\(A=\frac{x}{x-2}-\frac{2}{x+2}\)
\(=\frac{x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{2\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}\)
\(=\frac{x^2+2x-2x+4}{x^2-4}\)\(=\frac{x^2+4}{x^2-4}\)
b) \(A>0\) \(\Rightarrow\)\(\frac{x^2+4}{x^2-4}>0\)
Mà \(x^2+4>0\) \(\Rightarrow\)\(x^2-4>0\)
\(\Rightarrow\)\(x^2>4\)
Nếu x dương thì \(x>\sqrt{4}=2\)
Nếu x âm thì \(x< \sqrt{4}=2\)
\(\frac{\left(2x^3+2x\right)\left(x-2\right)^2}{\left(x^3-4x\right)\left(x+1\right)}\)
\(=\frac{2x\left(x^2+1\right)\left(x-2\right)^2}{x\left(x-2\right)\left(x+2\right)\left(x+1\right)}\)
\(=\frac{2\left(x^2+1\right)\left(x-2\right)}{\left(x+2\right)\left(x+1\right)}\)
Thay x=\(\frac{1}{2}\)
\(=\frac{2\left(\frac{1}{2}^2+1\right)\left(\frac{1}{2}-2\right)}{\left(\frac{1}{2}+2\right)\left(\frac{1}{2}+1\right)}\)
\(=-1\)