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

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

Solve for x

$$x=-7$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -2,-1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right)\left(x+2\right), the least common multiple of x^{2}+2x,x^{2}+x,x^{2}+3x+2.

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

Multiply x+1 and x+1 to get \left(x+1\right)^{2}.

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

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

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

Use the distributive property to multiply x+2 by x+4 and combine like terms.

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

To find the opposite of x^{2}+6x+8, find the opposite of each term.

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

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

$$2x+1-6x-8=x\left(-3\right)$$

Combine 2x and -6x to get -4x.

$$-4x+1-8=x\left(-3\right)$$

Subtract 8 from 1 to get -7.

$$-4x-7=x\left(-3\right)$$

Subtract x\left(-3\right) from both sides.

$$-4x-7-x\left(-3\right)=0$$

Combine -4x and -x\left(-3\right) to get -x.

$$-x-7=0$$

Add 7 to both sides. Anything plus zero gives itself.

$$-x=7$$

Multiply both sides by -1.

$$x=-7$$