Solve x^2+8x+11/x^2+5x+6+x-1/x+3=x-2/x+2

$$\frac { x ^ { 2 } + 8 x + 11 } { x ^ { 2 } + 5 x + 6 } + \frac { x - 1 } { x + 3 } = \frac { x - 2 } { x + 2 }$$

Solve for x

$$x=-5$$

Steps Using Factoring

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

$$x^{2}+8x+11+\left(x+2\right)\left(x-1\right)=\left(x+3\right)\left(x-2\right)$$

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

$$x^{2}+8x+11+x^{2}+x-2=\left(x+3\right)\left(x-2\right)$$

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

$$2x^{2}+8x+11+x-2=\left(x+3\right)\left(x-2\right)$$

Combine 8x and x to get 9x.

$$2x^{2}+9x+11-2=\left(x+3\right)\left(x-2\right)$$

Subtract 2 from 11 to get 9.

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

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

$$2x^{2}+9x+9=x^{2}+x-6$$

Subtract x^{2} from both sides.

$$2x^{2}+9x+9-x^{2}=x-6$$

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

$$x^{2}+9x+9=x-6$$

Subtract x from both sides.

$$x^{2}+9x+9-x=-6$$

Combine 9x and -x to get 8x.

$$x^{2}+8x+9=-6$$

Add 6 to both sides.

$$x^{2}+8x+9+6=0$$

Add 9 and 6 to get 15.

$$x^{2}+8x+15=0$$

To solve the equation, factor x^{2}+8x+15 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=8 ab=15$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 8.

$$a=3 b=5$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x+3\right)\left(x+5\right)$$

To find equation solutions, solve x+3=0 and x+5=0.

$$x=-3 x=-5$$

Variable x cannot be equal to -3.

$$x=-5$$

Steps Using Factoring By Grouping

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

$$x^{2}+8x+11+\left(x+2\right)\left(x-1\right)=\left(x+3\right)\left(x-2\right)$$

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

$$x^{2}+8x+11+x^{2}+x-2=\left(x+3\right)\left(x-2\right)$$

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

$$2x^{2}+8x+11+x-2=\left(x+3\right)\left(x-2\right)$$

Combine 8x and x to get 9x.

$$2x^{2}+9x+11-2=\left(x+3\right)\left(x-2\right)$$

Subtract 2 from 11 to get 9.

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

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

$$2x^{2}+9x+9=x^{2}+x-6$$

Subtract x^{2} from both sides.

$$2x^{2}+9x+9-x^{2}=x-6$$

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

$$x^{2}+9x+9=x-6$$

Subtract x from both sides.

$$x^{2}+9x+9-x=-6$$

Combine 9x and -x to get 8x.

$$x^{2}+8x+9=-6$$

Add 6 to both sides.

$$x^{2}+8x+9+6=0$$

Add 9 and 6 to get 15.

$$x^{2}+8x+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\times 15=15$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 8.

$$a=3 b=5$$

Rewrite x^{2}+8x+15 as \left(x^{2}+3x\right)+\left(5x+15\right).

$$\left(x^{2}+3x\right)+\left(5x+15\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(x+3\right)+5\left(x+3\right)$$

Factor out common term x+3 by using distributive property.

$$\left(x+3\right)\left(x+5\right)$$

To find equation solutions, solve x+3=0 and x+5=0.

$$x=-3 x=-5$$

Variable x cannot be equal to -3.

$$x=-5$$

Steps Using the Quadratic Formula

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

$$x^{2}+8x+11+\left(x+2\right)\left(x-1\right)=\left(x+3\right)\left(x-2\right)$$

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

$$x^{2}+8x+11+x^{2}+x-2=\left(x+3\right)\left(x-2\right)$$

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

$$2x^{2}+8x+11+x-2=\left(x+3\right)\left(x-2\right)$$

Combine 8x and x to get 9x.

$$2x^{2}+9x+11-2=\left(x+3\right)\left(x-2\right)$$

Subtract 2 from 11 to get 9.

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

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

$$2x^{2}+9x+9=x^{2}+x-6$$

Subtract x^{2} from both sides.

$$2x^{2}+9x+9-x^{2}=x-6$$

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

$$x^{2}+9x+9=x-6$$

Subtract x from both sides.

$$x^{2}+9x+9-x=-6$$

Combine 9x and -x to get 8x.

$$x^{2}+8x+9=-6$$

Add 6 to both sides.

$$x^{2}+8x+9+6=0$$

Add 9 and 6 to get 15.

$$x^{2}+8x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 15}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 15}}{2}$$

Multiply -4 times 15.

$$x=\frac{-8±\sqrt{64-60}}{2}$$

Add 64 to -60.

$$x=\frac{-8±\sqrt{4}}{2}$$

Take the square root of 4.

$$x=\frac{-8±2}{2}$$

Now solve the equation x=\frac{-8±2}{2} when ± is plus. Add -8 to 2.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

Now solve the equation x=\frac{-8±2}{2} when ± is minus. Subtract 2 from -8.

$$x=-\frac{10}{2}$$

Divide -10 by 2.

$$x=-5$$

The equation is now solved.

$$x=-3 x=-5$$

Variable x cannot be equal to -3.

$$x=-5$$

Steps for Completing the Square

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

$$x^{2}+8x+11+\left(x+2\right)\left(x-1\right)=\left(x+3\right)\left(x-2\right)$$

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

$$x^{2}+8x+11+x^{2}+x-2=\left(x+3\right)\left(x-2\right)$$

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

$$2x^{2}+8x+11+x-2=\left(x+3\right)\left(x-2\right)$$

Combine 8x and x to get 9x.

$$2x^{2}+9x+11-2=\left(x+3\right)\left(x-2\right)$$

Subtract 2 from 11 to get 9.

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

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

$$2x^{2}+9x+9=x^{2}+x-6$$

Subtract x^{2} from both sides.

$$2x^{2}+9x+9-x^{2}=x-6$$

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

$$x^{2}+9x+9=x-6$$

Subtract x from both sides.

$$x^{2}+9x+9-x=-6$$

Combine 9x and -x to get 8x.

$$x^{2}+8x+9=-6$$

Subtract 9 from both sides.

$$x^{2}+8x=-6-9$$

Subtract 9 from -6 to get -15.

$$x^{2}+8x=-15$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=-15+4^{2}$$

Square 4.

$$x^{2}+8x+16=-15+16$$

Add -15 to 16.

$$x^{2}+8x+16=1$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+4=1 x+4=-1$$

Subtract 4 from both sides of the equation.

$$x=-3 x=-5$$

Variable x cannot be equal to -3.

$$x=-5$$