Solve lambda-2/x+2+16/4-x^2=x+2/x

$$\frac { \lambda - 2 } { x + 2 } + \frac { 16 } { 4 - x ^ { 2 } } = \frac { x + 2 } { x }$$

Solve for λ

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

$$x\neq 0\text{ and }|x|\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(x-2\right)\left(x+2\right), the least common multiple of x+2,4-x^{2},x.

$$x\left(x-2\right)\left(\lambda -2\right)-x\times 16=\left(x^{2}-4\right)\left(x+2\right)$$

Use the distributive property to multiply x by x-2.

$$\left(x^{2}-2x\right)\left(\lambda -2\right)-x\times 16=\left(x^{2}-4\right)\left(x+2\right)$$

Use the distributive property to multiply x^{2}-2x by \lambda -2.

$$x^{2}\lambda -2x^{2}-2x\lambda +4x-x\times 16=\left(x^{2}-4\right)\left(x+2\right)$$

Multiply -1 and 16 to get -16.

$$x^{2}\lambda -2x^{2}-2x\lambda +4x-16x=\left(x^{2}-4\right)\left(x+2\right)$$

Combine 4x and -16x to get -12x.

$$x^{2}\lambda -2x^{2}-2x\lambda -12x=\left(x^{2}-4\right)\left(x+2\right)$$

Use the distributive property to multiply x^{2}-4 by x+2.

$$x^{2}\lambda -2x^{2}-2x\lambda -12x=x^{3}+2x^{2}-4x-8$$

Add 2x^{2} to both sides.

$$x^{2}\lambda -2x\lambda -12x=x^{3}+2x^{2}-4x-8+2x^{2}$$

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$x^{2}\lambda -2x\lambda -12x=x^{3}+4x^{2}-4x-8$$

Add 12x to both sides.

$$x^{2}\lambda -2x\lambda =x^{3}+4x^{2}-4x-8+12x$$

Combine -4x and 12x to get 8x.

$$x^{2}\lambda -2x\lambda =x^{3}+4x^{2}+8x-8$$

Combine all terms containing \lambda .

$$\left(x^{2}-2x\right)\lambda =x^{3}+4x^{2}+8x-8$$

Divide both sides by x^{2}-2x.

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

Dividing by x^{2}-2x undoes the multiplication by x^{2}-2x.

$$\lambda =\frac{x^{3}+4x^{2}+8x-8}{x^{2}-2x}$$

Divide 4x^{2}+8x-8+x^{3} by x^{2}-2x.

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