Solve 32/x^2+3x+2-3=x-3/x+1

$$\frac{ 32 }{ { x }^{ 2 } +3x+2 } -3 = \frac{ x-3 }{ x+1 }$$

Solve for x

$$x=-4$$

$$x=2$$

Steps Using Factoring By Grouping

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^{2}+3x+2,x+1.

$$32+\left(x+1\right)\left(x+2\right)\left(-3\right)=\left(x+2\right)\left(x-3\right)$$

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

$$32+\left(x^{2}+3x+2\right)\left(-3\right)=\left(x+2\right)\left(x-3\right)$$

Use the distributive property to multiply x^{2}+3x+2 by -3.

$$32-3x^{2}-9x-6=\left(x+2\right)\left(x-3\right)$$

Subtract 6 from 32 to get 26.

$$26-3x^{2}-9x=\left(x+2\right)\left(x-3\right)$$

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

$$26-3x^{2}-9x=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$26-3x^{2}-9x-x^{2}=-x-6$$

Combine -3x^{2} and -x^{2} to get -4x^{2}.

$$26-4x^{2}-9x=-x-6$$

Add x to both sides.

$$26-4x^{2}-9x+x=-6$$

Combine -9x and x to get -8x.

$$26-4x^{2}-8x=-6$$

Add 6 to both sides.

$$26-4x^{2}-8x+6=0$$

Add 26 and 6 to get 32.

$$32-4x^{2}-8x=0$$

Divide both sides by 4.

$$8-x^{2}-2x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-8=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -2.

$$a=2 b=-4$$

Rewrite -x^{2}-2x+8 as \left(-x^{2}+2x\right)+\left(-4x+8\right).

$$\left(-x^{2}+2x\right)+\left(-4x+8\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(-x+2\right)+4\left(-x+2\right)$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(x+4\right)$$

To find equation solutions, solve -x+2=0 and x+4=0.

$$x=2 x=-4$$

Steps Using the Quadratic Formula

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^{2}+3x+2,x+1.

$$32+\left(x+1\right)\left(x+2\right)\left(-3\right)=\left(x+2\right)\left(x-3\right)$$

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

$$32+\left(x^{2}+3x+2\right)\left(-3\right)=\left(x+2\right)\left(x-3\right)$$

Use the distributive property to multiply x^{2}+3x+2 by -3.

$$32-3x^{2}-9x-6=\left(x+2\right)\left(x-3\right)$$

Subtract 6 from 32 to get 26.

$$26-3x^{2}-9x=\left(x+2\right)\left(x-3\right)$$

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

$$26-3x^{2}-9x=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$26-3x^{2}-9x-x^{2}=-x-6$$

Combine -3x^{2} and -x^{2} to get -4x^{2}.

$$26-4x^{2}-9x=-x-6$$

Add x to both sides.

$$26-4x^{2}-9x+x=-6$$

Combine -9x and x to get -8x.

$$26-4x^{2}-8x=-6$$

Add 6 to both sides.

$$26-4x^{2}-8x+6=0$$

Add 26 and 6 to get 32.

$$32-4x^{2}-8x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-4x^{2}-8x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -8 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-4\right)\times 32}}{2\left(-4\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-4\right)\times 32}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-8\right)±\sqrt{64+16\times 32}}{2\left(-4\right)}$$

Multiply 16 times 32.

$$x=\frac{-\left(-8\right)±\sqrt{64+512}}{2\left(-4\right)}$$

Add 64 to 512.

$$x=\frac{-\left(-8\right)±\sqrt{576}}{2\left(-4\right)}$$

Take the square root of 576.

$$x=\frac{-\left(-8\right)±24}{2\left(-4\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±24}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{8±24}{-8}$$

Now solve the equation x=\frac{8±24}{-8} when ± is plus. Add 8 to 24.

$$x=\frac{32}{-8}$$

Divide 32 by -8.

$$x=-4$$

Now solve the equation x=\frac{8±24}{-8} when ± is minus. Subtract 24 from 8.

$$x=-\frac{16}{-8}$$

Divide -16 by -8.

$$x=2$$

The equation is now solved.

$$x=-4 x=2$$

Steps for Completing the Square

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^{2}+3x+2,x+1.

$$32+\left(x+1\right)\left(x+2\right)\left(-3\right)=\left(x+2\right)\left(x-3\right)$$

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

$$32+\left(x^{2}+3x+2\right)\left(-3\right)=\left(x+2\right)\left(x-3\right)$$

Use the distributive property to multiply x^{2}+3x+2 by -3.

$$32-3x^{2}-9x-6=\left(x+2\right)\left(x-3\right)$$

Subtract 6 from 32 to get 26.

$$26-3x^{2}-9x=\left(x+2\right)\left(x-3\right)$$

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

$$26-3x^{2}-9x=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$26-3x^{2}-9x-x^{2}=-x-6$$

Combine -3x^{2} and -x^{2} to get -4x^{2}.

$$26-4x^{2}-9x=-x-6$$

Add x to both sides.

$$26-4x^{2}-9x+x=-6$$

Combine -9x and x to get -8x.

$$26-4x^{2}-8x=-6$$

Subtract 26 from both sides.

$$-4x^{2}-8x=-6-26$$

Subtract 26 from -6 to get -32.

$$-4x^{2}-8x=-32$$

Divide both sides by -4.

$$\frac{-4x^{2}-8x}{-4}=-\frac{32}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{8}{-4}\right)x=-\frac{32}{-4}$$

Divide -8 by -4.

$$x^{2}+2x=-\frac{32}{-4}$$

Divide -32 by -4.

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

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square 1.

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

Add 8 to 1.

$$x^{2}+2x+1=9$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{9}$$

Simplify.

$$x+1=3 x+1=-3$$

Subtract 1 from both sides of the equation.

$$x=2 x=-4$$