Solve E(x)=(x+3/x+1-x+1/x+3

$$E ( x ) = ( \frac { x + 3 } { x + 1 } - \frac { x + 1 } { x + 3 }$$

Solve for E

$$E=\frac{4\left(x+2\right)}{x\left(x+1\right)\left(x+3\right)}$$

$$x\neq -3\text{ and }x\neq -1\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+1\right)\left(x+3\right), the least common multiple of x+1,x+3.

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

Multiply x+3 and x+3 to get \left(x+3\right)^{2}.

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

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

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

Use the distributive property to multiply Ex by x+1.

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

Use the distributive property to multiply Ex^{2}+Ex by x+3 and combine like terms.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$Ex^{3}+4Ex^{2}+3Ex=x^{2}+6x+9-\left(x+1\right)^{2}$$

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

$$Ex^{3}+4Ex^{2}+3Ex=x^{2}+6x+9-\left(x^{2}+2x+1\right)$$

To find the opposite of x^{2}+2x+1, find the opposite of each term.

$$Ex^{3}+4Ex^{2}+3Ex=x^{2}+6x+9-x^{2}-2x-1$$

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

$$Ex^{3}+4Ex^{2}+3Ex=6x+9-2x-1$$

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

$$Ex^{3}+4Ex^{2}+3Ex=4x+9-1$$

Subtract 1 from 9 to get 8.

$$Ex^{3}+4Ex^{2}+3Ex=4x+8$$

Combine all terms containing E.

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

Divide both sides by x^{3}+4x^{2}+3x.

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

Dividing by x^{3}+4x^{2}+3x undoes the multiplication by x^{3}+4x^{2}+3x.

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

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

$$E=\frac{4\left(x+2\right)}{x\left(x+1\right)\left(x+3\right)}$$