Solve x+6/x+1-3x^2+10x+4/x^2+3x+2+2x+1/x+2=0

$$\frac { x + 6 } { x + 1 } - \frac { 3 x ^ { 2 } + 10 x + 4 } { x ^ { 2 } + 3 x + 2 } + \frac { 2 x + 1 } { x + 2 } = 0$$

Solve for x

$$x=-9$$

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}+3x+2,x+2.

$$\left(x+2\right)\left(x+6\right)-\left(3x^{2}+10x+4\right)+\left(x+1\right)\left(2x+1\right)=0$$

Use the distributive property to multiply x+2 by x+6 and combine like terms.

$$x^{2}+8x+12-\left(3x^{2}+10x+4\right)+\left(x+1\right)\left(2x+1\right)=0$$

To find the opposite of 3x^{2}+10x+4, find the opposite of each term.

$$x^{2}+8x+12-3x^{2}-10x-4+\left(x+1\right)\left(2x+1\right)=0$$

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

$$-2x^{2}+8x+12-10x-4+\left(x+1\right)\left(2x+1\right)=0$$

Combine 8x and -10x to get -2x.

$$-2x^{2}-2x+12-4+\left(x+1\right)\left(2x+1\right)=0$$

Subtract 4 from 12 to get 8.

$$-2x^{2}-2x+8+\left(x+1\right)\left(2x+1\right)=0$$

Use the distributive property to multiply x+1 by 2x+1 and combine like terms.

$$-2x^{2}-2x+8+2x^{2}+3x+1=0$$

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

$$-2x+8+3x+1=0$$

Combine -2x and 3x to get x.

$$x+8+1=0$$

Add 8 and 1 to get 9.

$$x+9=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$x=-9$$