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`\text{Đáp án và giải thích các bước giải:}`

`\text{Bài 1:}`

`a)`  `\frac{4}{x+2}+\frac{3}{x-2}+\frac{5x+2}{4-x^2}`

`=\frac{4}{x+2}+\frac{3}{x-2}-\frac{5x+2}{x^2-4}`

`=\frac{4(x-2)}{(x-2)(x+2)}+\frac{3(x+2)}{(x-2)(x+2)}-\frac{5x+2}{(x-2)(x+2)}`

`=\frac{4(x-2)+3(x+2)-5x-2}{(x-2)(x+2)}`

`=\frac{4x-8+3x+6-5x-2}{x^2-4}`

`=\frac{2x-2}{x^2-4}`

`b)`  `\frac{1-2x}{2x}+\frac{2x}{2x-1}+\frac{1}{2x-4x^2}`

`=\frac{1-2x}{2x}+\frac{2x}{2x-1}-\frac{1}{4x^2-2x}`

`=\frac{(1-2x)(2x-1)}{2x(2x-1)}+\frac{2x.2x}{2x(2x-1)}-\frac{1}{2x(2x-1)}`

`=\frac{(1-2x)(2x-1)+2x.2x-1}{2x(2x-1)}`

`=\frac{(1-2x)(2x-1)+4x^2-1}{2x(2x-1)}`

`=\frac{(1-2x)(2x-1)+(2x-1)(2x+1){2x(2x-1)}`

`=\frac{(2x-1)(1-2x+2x+1)}{2x(2x-1)}`

`=\frac{2(2x-1)}{2x(2x-1)}`

`=\frac{1}{x}`

`c)`  `\frac{1}{x-y}+\frac{3xy}{y^3-x^3}+\frac{x-y}{x^2+xy+y^2}`

`=\frac{1}{x-y}-\frac{3xy}{(x^3-y^3}+\frac{x-y}{x^2+xy+y^2}`

`=\frac{x^2+xy+y^2}{(x-y)(x^2+xy+y^2)}-\frac{3xy}{(x-y)(x^2+xy+y^2)}+\frac{(x-y)(x-y)}{(x-y)(x^2+xy+y^2)}`

`=\frac{x^2+xy+y^2-3xy+(x-y)^2}{(x-y)(x^2+xy+y^2)}`

`=\frac{x^2+xy+y^2-3xy+x^2-2xy+y^2}{(x-y)(x^2+xy+y^2)}`

`=\frac{2x^2-4xy+2y^2}{(x-y)(x^2+xy+y^2)}`

`=\frac{2x^2-2xy-2xy+2y^2}{(x-y)(x^2+xy+y^2)}`

`=\frac{2x(x-y)-2y(x-y)}{(x-y)(x^2+xy+y^2)}`

`=\frac{(x-y)(2x-2y)}{(x-y)(x^2+xy+y^2)}`

`=\frac{2x-2y}{(x-y)(x^2+xy+y^2)}`