1) \(\frac{x+1}{2x-2}+\frac{x^2+3}{2-2x^2}\)

\(=\frac{-4x^2+8x-4}{-4x^3+4x^2+4x-4}\)

\(=\frac{-x^2+2x-1}{-x^3+x^2+x-1}\)

\(=\frac{\left(-x+1\right)\left(x-1\right)}{\left(-x-1\right)\left(x-1\right)\left(x-1\right)}\)

\(=\frac{1}{x+1}\)

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

\(=\frac{-16x^3+16x^2-4x}{-16x^4+16x^3-4x^2}\)

\(=\frac{-16x^2+16x-4}{-16x^3+16x^2-4x}\)

\(=\frac{-4x^2+4x-1}{-4x^3+4x^2-x}\)

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

\(=\frac{1}{x}\)

ĐKXĐ : \(x\ne\pm\frac{1}{2}\)

\(E=\left(\frac{\left(4x^2+2x\right)\left(1+4x^2\right)}{\left(1-4x^2\right)\left(1+4x^2\right)}-\frac{\left(4x^2-2x\right)\left(1-4x^2\right)}{\left(1-4x^2\right)\left(1+4x^2\right)}\right):\left(\frac{\left(1+2x\right)\left(1+2x\right)}{\left(1-2x\right)\left(1+2x\right)}-\frac{\left(1-2x\right)\left(1-2x\right)}{\left(1+2x\right)\left(1-2x\right)}\right)\)

\(E=\left(\frac{16x^4+8x^3+4x^2+2x+16x^4-8x^3-4x^2+2x}{1-16x^4}\right):\left(\frac{1+2x+x^2-1+2x-x^2}{1-4x^2}\right)\)

\(E=\frac{32x^4+4x}{1-16x^4}:\frac{4x}{1-4x^2}\)

\(E=\frac{4x\left(8x^3+1\right)}{\left(1-4x^2\right)\left(1+4x^2\right)}.\frac{1-4x^2}{4x}\)

\(E=\frac{8x^3+1}{1+4x^2}\)

Study well 

E=\(\left(\frac{4x^2+2x}{1-4x^2}-\frac{4x^2-2x}{1+4x^2}\right):\left(\frac{1+2x}{1-2x}-\frac{1-2x}{1+2x}\right)\)

E=\(\left(\frac{\left(4x^2+2x\right)\left(1+4x^2\right)-\left(4x^2-2x\right)\left(1-4x^2\right)}{\left(1-4x^2\right)\left(1+4x^2\right)}\right):\)\(\left(\frac{\left(1+2x\right)^2-\left(1-2x\right)^2}{\left(1-2x\right)\left(1+2x\right)}\right)\)

E=\(\frac{4x^2+16x^4+2x+8x^3-\left(4x^2-16x^4-2x+8x^3\right)}{\left(1-4x^2\right)\left(1+4x^2\right)}:\)\(\left(\frac{\left(1+4x+4x^2\right)-\left(1-4x+4x^2\right)}{\left(1-2x\right)\left(1+2x\right)}\right)\)

E=\(\frac{4x^2+16x^4+2x+8x^3-4x^2+16x^4+2x-8x^3}{\left(1-4x^2\right)\left(1+4x^2\right)}:\)\(\left(\frac{1+4x+4x^2-1+4x-4x^2}{\left(1-2x\right)\left(1+2x\right)}\right)\)

E=\(\frac{16x^4+2x+16x^4+2x}{\left(1-4x^2\right)\left(1+4x^2\right)}:\)\(\left(\frac{8x}{\left(1-2x\right)\left(1+2x\right)}\right)\)

E=\(\frac{32x^4+8x}{\left(1-4x^2\right)\left(1+4x^2\right)}.\frac{1-4x^2}{8x}\)

E=\(\frac{8x\left(4x^3+1\right)}{\left(1-4x^2\right)\left(1+4x^2\right)}.\frac{1-4x^2}{8x}\)

E=\(\frac{4x^3+1}{1+4x^2}\)

Góp ý : 

Cô giáo ơi

Cô tính nhầm rồi 2x + 2x = 4x cô ạ chứ không phải 8x đâu cô

mong cô xem lại ạ 

\(=\dfrac{4x\left(x+1\right)+1}{4x^2}\cdot\left(\dfrac{-\left(2x-1\right)}{2x+1}+\dfrac{1}{\left(2x+1\right)\left(2x-1\right)}\cdot\dfrac{\left(2x-1\right)^2}{2x+1}\right)-\dfrac{1}{2x}\)

\(=\dfrac{\left(2x+1\right)^2}{4x^2}\cdot\left(\dfrac{-\left(2x-1\right)}{2x+1}+\dfrac{2x-1}{\left(2x+1\right)^2}\right)-\dfrac{1}{2x}\)

\(=\dfrac{\left(2x+1\right)^2}{4x^2}\cdot\dfrac{-\left(2x-1\right)\left(2x+1\right)+2x-1}{\left(2x+1\right)^2}-\dfrac{1}{2x}\)

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

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

\(=\dfrac{-2x\left(2x-1\right)}{2x\cdot2x}-\dfrac{1}{2x}\)

\(=\dfrac{-2x+1-1}{2x}=\dfrac{-2x}{2x}=-1\)

E=\(\frac{\left(4x^2+2x\right)\left(1+4x^2\right)-\left(4x^2-2x\right)\left(1-4x^2\right)}{\left(1-4x^2\right)\left(1+4x^2\right)}:\frac{\left(1+2x\right)^2-\left(1-2x\right)^2}{1-4x^2}\)

E=\(\frac{4x^2+16x^4+2x+8x^3-4x^2+16x^2+2x-8x^3}{\left(1-4x^2\right)\left(1+4x^2\right)}.\frac{1-4x^2}{1+4x+4x^2-1+4x-4x^2}\)

E=\(\frac{32x^4+4x}{8x\left(1+4x^2\right)}=\frac{8x^3+1}{2\left(1+4x^2\right)}\)

ĐKXĐ: \(x\ne0;x\ne\frac{1}{2}\)

\(\frac{2x^2+1}{4x^2-2x}+\frac{3}{2x}-\frac{3-3x}{2x-1}\)

\(=\frac{2x^2+1}{4x^2-2x}+\frac{3}{2x}-\frac{6x-6x^2}{4x^2-2x}\)

\(=\frac{8x^2-6x+1}{4x^2-2x}+\frac{3}{2x}=\frac{8\left(x-\frac{1}{2}\right)\left(x-\frac{1}{4}\right)}{4x\left(x-\frac{1}{2}\right)}+\frac{3}{2x}\)

\(=\frac{8x-2}{4x}+\frac{3}{2x}=\frac{8x-2}{4x}+\frac{6}{4x}=\frac{8x-2+6}{4x}\)

\(=\frac{8x+4}{4x}=1+\frac{4x+4}{4x}=2+\frac{4}{4x}=2+\frac{1}{x}\)

Ơ bài t có gì sai????lại là bọn dis dạo nữa cơ à? Ok,ok cho chúng m dis,t cx méo quan tâm.Và t biết bài t đúng!

Đề là rút gọn? Điều kiện: x khác o và x khác 1/2

\(\frac{1-2x}{2x}+\frac{2x}{2x-1}+\frac{1}{2x\left(1-2x\right)}=\frac{\left(1-2X\right)\left(2x-1\right)}{2x.\left(2x-1\right)}+\frac{2x.2x}{\left(2x-1\right).2x}-\frac{1}{2x\left(2x-1\right)}=\)

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

ĐKXĐ: \(\left\{{}\begin{matrix}x\ne\frac{1}{2}\\x\ne-\frac{1}{2}\\x\ne0\end{matrix}\right.\)

\(\left(\frac{2x+1}{2x-1}-\frac{2x-1}{2x+1}\right):\left[1:\left(1-\frac{1}{x}+\frac{1}{4x^2}\right)\right]\)

\(=\left[\frac{\left(2x+1\right)^2-\left(2x-1\right)^2}{\left(2x-1\right)\left(2x+1\right)}\right]:\left[1:\frac{4x^2-4x+1}{4x^2}\right]\)

\(=\frac{4x^2+4x+1-4x^2+4x-1}{\left(2x-1\right)\left(2x+1\right)}:\frac{4x^2}{\left(2x-1\right)^2}\)

\(=\frac{8x}{\left(2x-1\right)\left(2x+1\right)}.\frac{\left(2x-1\right)^2}{4x^2}=\frac{2\left(2x-1\right)}{\left(2x+1\right).x}=\frac{4x-2}{2x^2+x}\left(ĐPCM\right)\)

\(\frac{3}{2x}+\frac{3x-3}{2x-1}+\frac{2x^2+1}{4x^2-2x}\)

= \(\frac{3\left(2x-1\right)+2x\left(3x-3\right)+2x^2+1}{4x^2-2x}\)

= \(\frac{6x-3+6x^2-6x+2x^2+1}{4x^2-2x}\)

= \(\frac{8x^2-2}{4x^2-2x}\)

= \(\frac{2}{2x}\)