a: \(B=\frac{4x}{x+1}+\frac{x}{1-x}+\frac{2x}{x^2-1}\)
\(=\frac{4x}{x+1}-\frac{x}{x-1}+\frac{2x}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{4x\left(x-1\right)-x\left(x+1\right)+2x}{\left(x-1\right)\left(x+1\right)}=\frac{4x^2-4x-x^2-x+2x}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{3x^2-3x}{\left(x-1\right)\left(x+1\right)}=\frac{3x\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}=\frac{3x}{x+1}\)
\(P=A\cdot B=\frac{3x}{x+1}\cdot\frac{x-2}{x}=\frac{3\left(x-2\right)}{x+1}=\frac{3x-6}{x+1}\)
b: Để P là số tự nhiên thì P>=0 và 3x-6⋮x+1
=>3x+3-9⋮x+1 và \(\frac{3x-6}{x+1}\ge0\)
=>-9⋮x+1 và \(\frac{x-2}{x+1}\ge0\)
=>x+1∈{1;-1;3;-3;9;-9} và \(\left[\begin{array}{l}x\ge2\\ x<-1\end{array}\right.\)
=>x∈{0;-2;2;-4;8;-10} và \(\left[\begin{array}{l}x\ge2\\ x<-1\end{array}\right.\)
=>x∈{2;-2;-4;8;-10}
