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

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

Solve for x

$$x=-\frac{1}{2}=-0.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)^{2}, the least common multiple of x+1,\left(x+1\right)^{2}.

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

Consider \left(x+1\right)\left(x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$x^{2}-1=2x+x^{2}+2x+1$$

Combine 2x and 2x to get 4x.

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

Subtract 4x from both sides.

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

Subtract x^{2} from both sides.

$$x^{2}-1-4x-x^{2}=1$$

Combine x^{2} and -x^{2} to get 0.

$$-1-4x=1$$

Add 1 to both sides.

$$-4x=1+1$$

Add 1 and 1 to get 2.

$$-4x=2$$

Divide both sides by -4.

$$x=\frac{2}{-4}$$

Reduce the fraction \frac{2}{-4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{2}$$