Solve left(x+1right)left(2-2left(x+1right)/1-xright)=2x*-2left(x+1right)/1-x

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

Solve for x

$$x\neq 1$$

Short Solution Steps

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by -x+1.

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

Use the distributive property to multiply 2 by x+1.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{1-x}{1-x}.

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

Since \frac{2\left(1-x\right)}{1-x} and \frac{2x+2}{1-x} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(1-x\right)-\left(2x+2\right).

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

Combine like terms in 2-2x-2x-2.

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

Express \left(x+1\right)\times \frac{-4x}{1-x} as a single fraction.

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

Express \frac{\left(x+1\right)\left(-4\right)x}{1-x}\left(-x+1\right) as a single fraction.

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

Multiply 2 and -2 to get -4.

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

Use the distributive property to multiply -4x by x+1.

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

Use the distributive property to multiply x+1 by -4.

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

Use the distributive property to multiply -4x-4 by x.

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

Use the distributive property to multiply -4x^{2}-4x by -x+1 and combine like terms.

$$\frac{4x^{3}-4x}{1-x}=-4x^{2}-4x$$

Add 4x^{2} to both sides.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 4x^{2} times \frac{1-x}{1-x}.

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

Since \frac{4x^{3}-4x}{1-x} and \frac{4x^{2}\left(1-x\right)}{1-x} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 4x^{3}-4x+4x^{2}\left(1-x\right).

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

Combine like terms in 4x^{3}-4x+4x^{2}-4x^{3}.

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

Add 4x to both sides.

$$\frac{-4x+4x^{2}}{1-x}+4x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4x times \frac{1-x}{1-x}.

$$\frac{-4x+4x^{2}}{1-x}+\frac{4x\left(1-x\right)}{1-x}=0$$

Since \frac{-4x+4x^{2}}{1-x} and \frac{4x\left(1-x\right)}{1-x} have the same denominator, add them by adding their numerators.

$$\frac{-4x+4x^{2}+4x\left(1-x\right)}{1-x}=0$$

Do the multiplications in -4x+4x^{2}+4x\left(1-x\right).

$$\frac{-4x+4x^{2}+4x-4x^{2}}{1-x}=0$$

Combine like terms in -4x+4x^{2}+4x-4x^{2}.

$$\frac{0}{1-x}=0$$

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by -x+1.

$$0=0$$

This is true for any x.

$$x\in \mathrm{R}$$

Variable x cannot be equal to 1.

$$x\in \mathrm{R}\setminus 1$$