Solve 2x/x-1+x(x^2+2)/x^2+x-2=3x+4/x+2+x^2/x+2

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

Solve for x

$$x=-\frac{4}{5}=-0.8$$

Steps for Solving Linear Equation

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

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

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

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

Use the distributive property to multiply 2x+4 by x.

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

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

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

Combine 4x and 2x to get 6x.

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

Use the distributive property to multiply x-1 by 3x+4 and combine like terms.

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

Use the distributive property to multiply x-1 by x^{2}.

$$2x^{2}+6x+x^{3}=3x^{2}+x-4+x^{3}-x^{2}$$

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$2x^{2}+6x+x^{3}=2x^{2}+x-4+x^{3}$$

Subtract 2x^{2} from both sides.

$$2x^{2}+6x+x^{3}-2x^{2}=x-4+x^{3}$$

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

$$6x+x^{3}=x-4+x^{3}$$

Subtract x from both sides.

$$6x+x^{3}-x=-4+x^{3}$$

Combine 6x and -x to get 5x.

$$5x+x^{3}=-4+x^{3}$$

Subtract x^{3} from both sides.

$$5x+x^{3}-x^{3}=-4$$

Combine x^{3} and -x^{3} to get 0.

$$5x=-4$$

Divide both sides by 5.

$$x=\frac{-4}{5}$$

Fraction \frac{-4}{5} can be rewritten as -\frac{4}{5} by extracting the negative sign.

$$x=-\frac{4}{5}$$