\(a,\dfrac{a^2}{a+1}+\dfrac{1}{a-1}-\dfrac{2a}{a^2-1}\\ =\dfrac{a^2}{a+1}+\dfrac{1}{a-1}-\dfrac{2a}{\left(a-1\right)\left(a+1\right)}\\ =\dfrac{a^2\left(a-1\right)}{\left(a+1\right)\left(a-1\right)}+\dfrac{a+1}{\left(a-1\right)\left(a+1\right)}-\dfrac{2a}{\left(a-1\right)\left(a+1\right)}\\ =\dfrac{a^3-a^2+a+1-2a}{\left(a-1\right)\left(a+1\right)}\\ =\dfrac{a^3-a^2-a+1}{\left(a-1\right)\left(a+1\right)}\\ =\dfrac{\left(a-1\right)^2\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}\\ =a-1\)
\(b,\dfrac{4}{x+2}+\dfrac{3}{2-x}+\dfrac{12}{x^2-4}\\ =\dfrac{4}{x+2}-\dfrac{3}{x-2}+\dfrac{12}{\left(x-2\right)\left(x+2\right)}\\ =\dfrac{4\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}-\dfrac{3\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\dfrac{12}{\left(x-2\right)\left(x+2\right)}\\ =\dfrac{4x-8-3x-6+12}{\left(x-2\right)\left(x+2\right)}\\ =\dfrac{x-2}{\left(x-2\right)\left(x+2\right)}\\ =\dfrac{1}{x+2}\)
\(c,\dfrac{4}{x+2}+\dfrac{2}{x-2}+\dfrac{5x-6}{4-x^2}\\ =\dfrac{4}{2+x}-\dfrac{2}{2-x}+\dfrac{5x-6}{\left(2-x\right)\left(2+x\right)}\\ =\dfrac{4\left(2-x\right)}{\left(2+x\right)\left(2-x\right)}-\dfrac{2\left(2+x\right)}{\left(2-x\right)\left(2+x\right)}+\dfrac{5x-6}{\left(2-x\right)\left(2+x\right)}\\ =\dfrac{8-4x-4-2x+5x-6}{\left(2-x\right)\left(2+x\right)}\\ =\dfrac{-x-2}{\left(2-x\right)\left(2+x\right)}\\ =\dfrac{-\left(x+2\right)}{\left(2-x\right)\left(2+x\right)}\\ =-\dfrac{1}{2-x}\)
\(h,\dfrac{7}{x}-\dfrac{x}{x+6}+\dfrac{36}{x^2+6x}\\ =\dfrac{7}{x}-\dfrac{x}{x+6}+\dfrac{36}{x\left(x+6\right)}\\ =\dfrac{7\left(x+6\right)}{x\left(x+6\right)}-\dfrac{x^2}{\left(x+6\right)x}+\dfrac{36}{x\left(x+6\right)}\\ =\dfrac{7x+42-x^2+36}{x\left(x+6\right)}\\ =\dfrac{-x^2+7x+78}{x\left(x+6\right)}\\ =\dfrac{-\left(x-13\right)\left(x+6\right)}{x\left(x+6\right)}\\ =\dfrac{-x+13}{x}\)
\(i,\dfrac{3}{2x}+\dfrac{3x-3}{2x-1}+\dfrac{2x^2+1}{4x^2-2x}\\ =\dfrac{3}{2x}+\dfrac{3x-3}{2x-1}+\dfrac{2x^2+1}{2x\left(2x-1\right)}\\ =\dfrac{3\left(2x-1\right)}{2x\left(2x-1\right)}+\dfrac{2x\left(3x-3\right)}{2x\left(2x-1\right)}+\dfrac{2x^2+1}{2x\left(2x-1\right)}\\ =\dfrac{6x-3+6x^2-6x+2x^2+1}{2x\left(2x-1\right)}\\ =\dfrac{8x^2-2}{2x\left(2x-1\right)}\\ =\dfrac{2\left(4x^2-1\right)}{2x\left(2x-1\right)}\\ =\dfrac{2\left(2x-1\right)\left(2x+1\right)}{2x\left(2x-1\right)}\\ =\dfrac{2x+1}{x}\)
\(k,\dfrac{1-3x}{2x}+\dfrac{3x-2}{2x-1}+\dfrac{3x-2}{2x-4x^2}\\ =\dfrac{1-3x}{2x}-\dfrac{3x-2}{1-2x}+\dfrac{3x-2}{2x\left(1-2x\right)}\\ =\dfrac{\left(1-3x\right)\left(1-2x\right)}{2x\left(1-2x\right)}-\dfrac{2x\left(3x-2\right)}{\left(1-2x\right)2x}+\dfrac{3x-2}{2x\left(1-2x\right)}\\ =\dfrac{1-2x-3x+6x^2-6x^2+4x+3x-2}{2x\left(1-2x\right)}\\ =\dfrac{2x-1}{2x\left(1-2x\right)}\\ =\dfrac{2x-1}{-2x\left(2x-1\right)}\\ =-\dfrac{1}{2x}\)
