BÀi 1:
a: \(\frac{2x+5}{3}+\frac{x-2}{3}\)
\(=\frac{2x+5+x-2}{3}=\frac{3x+3}{3}\)
=x+1
b: \(\frac{a}{a+b}+\frac{b}{b+a}=\frac{a+b}{a+b}=1\)
c: \(\frac{a}{a-1}+\frac{1}{1-a}=\frac{a-1}{a-1}=1\)
Bài 2:
a: \(\frac{x}{x-y}+\frac{x}{x+y}+\frac{2y^2}{x^2-y^2}\)
\(=\frac{x\left(x+y\right)+x\left(x-y\right)+2y^2}{\left(x-y\right)\left(x+y\right)}\)
\(=\frac{x^2+xy+x^2-xy+2y^2}{\left(x-y\right)\left(x+y\right)}=\frac{2x^2+2y^2}{x^2-y^2}\)
b: \(\frac{1}{x+2}+\frac{1}{2-x}+\frac{2x^2}{x^2-4}\)
\(=\frac{1}{x+2}-\frac{1}{x-2}+\frac{2x^2}{\left(x-2\right)\left(x+2\right)}\)
\(=\frac{x-2-x-2+2x^2}{\left(x-2\right)\left(x+2\right)}=\frac{2x^2-4}{x^2-4}\)
c: \(\frac{x^2+2}{x^3-1}+\frac{x}{x^2+x+1}+\frac{1}{1-x}\)
\(=\frac{x^2+2}{\left(x-1\right)\left(x^2+x+1\right)}+\frac{x}{x^2+x+1}-\frac{1}{x-1}\)
\(=\frac{x^2+2+x\left(x-1\right)-x^2-x-1}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{-x+1+x^2-x}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\frac{\left(x-1\right)^2}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{x-1}{x_{}^2+x+1}\)
d: \(\frac{x^2+3x}{x^2+6x+9}+\frac{3}{x-3}+\frac{6x}{9-x^2}\)
\(=\frac{x\left(x+3\right)}{\left(x+3\right)^2}+\frac{3}{x-3}-\frac{6x}{\left(x-3\right)\left(x+3\right)}\)
\(=\frac{x}{x+3}+\frac{3}{x-3}-\frac{6x}{\left(x-3\right)\left(x+3\right)}=\frac{x\left(x-3\right)+3\left(x+3\right)-6x}{\left(x-3\right)\left(x+3\right)}\)
\(=\frac{x^2-6x+9}{\left(x-3\right)\left(x+3\right)}=\frac{\left(x-3\right)^2}{\left(x-3\right)\left(x+3\right)}=\frac{x-3}{x+3}\)
