Solve 2x+3/x^2+5x+6+3x-2/x^2+x-6=5x-2/x^2-4

$$\frac { 2 x + 3 } { x ^ { 2 } + 5 x + 6 } + \frac { 3 x - 2 } { x ^ { 2 } + x - 6 } = \frac { 5 x - 2 } { x ^ { 2 } - 4 }$$

Solve for x

$$x=-\frac{2}{5}=-0.4$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -3,-2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right)\left(x+3\right), the least common multiple of x^{2}+5x+6,x^{2}+x-6,x^{2}-4.

$$\left(x-2\right)\left(2x+3\right)+\left(x+2\right)\left(3x-2\right)=\left(x+3\right)\left(5x-2\right)$$

Use the distributive property to multiply x-2 by 2x+3 and combine like terms.

$$2x^{2}-x-6+\left(x+2\right)\left(3x-2\right)=\left(x+3\right)\left(5x-2\right)$$

Use the distributive property to multiply x+2 by 3x-2 and combine like terms.

$$2x^{2}-x-6+3x^{2}+4x-4=\left(x+3\right)\left(5x-2\right)$$

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$5x^{2}-x-6+4x-4=\left(x+3\right)\left(5x-2\right)$$

Combine -x and 4x to get 3x.

$$5x^{2}+3x-6-4=\left(x+3\right)\left(5x-2\right)$$

Subtract 4 from -6 to get -10.

$$5x^{2}+3x-10=\left(x+3\right)\left(5x-2\right)$$

Use the distributive property to multiply x+3 by 5x-2 and combine like terms.

$$5x^{2}+3x-10=5x^{2}+13x-6$$

Subtract 5x^{2} from both sides.

$$5x^{2}+3x-10-5x^{2}=13x-6$$

Combine 5x^{2} and -5x^{2} to get 0.

$$3x-10=13x-6$$

Subtract 13x from both sides.

$$3x-10-13x=-6$$

Combine 3x and -13x to get -10x.

$$-10x-10=-6$$

Add 10 to both sides.

$$-10x=-6+10$$

Add -6 and 10 to get 4.

$$-10x=4$$

Divide both sides by -10.

$$x=\frac{4}{-10}$$

Reduce the fraction \frac{4}{-10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{5}$$