Solve x^2-x-2/(x+3)^2(x-2)*x/x+1=0

$$\frac { x ^ { 2 } - x - 2 } { ( x + 3 ) ^ { 2 } ( x - 2 ) } \cdot \frac { x } { x + 1 } = 0$$

Solve for x

$$x=0$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,-1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+1\right)\left(x+3\right)^{2}, the least common multiple of \left(x+3\right)^{2}\left(x-2\right),x+1.

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

Express \left(x+1\right)\times \frac{x}{x+1} as a single fraction.

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

Use the distributive property to multiply \frac{\left(x+1\right)x}{x+1} by x^{2}-x-2.

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

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

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

Express \frac{x^{2}+x}{x+1}x^{2} as a single fraction.

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

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

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

Express \frac{x^{2}+x}{x+1}x as a single fraction.

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

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

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

Express -2\times \frac{x^{2}+x}{x+1} as a single fraction.

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

Since \frac{\left(x^{2}+x\right)x^{2}}{x+1} and \frac{\left(x^{2}+x\right)x}{x+1} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{2}+x\right)x^{2}-\left(x^{2}+x\right)x.

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

Combine like terms in x^{4}+x^{3}-x^{3}-x^{2}.

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

Since \frac{x^{4}-x^{2}}{x+1} and \frac{-2\left(x^{2}+x\right)}{x+1} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x^{4}-x^{2}-2\left(x^{2}+x\right).

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

Combine like terms in x^{4}-x^{2}-2x^{2}-2x.

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

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$x^{4}-3x^{2}-2x=0$$

Substitute t for x^{2}.

$$t^{2}-3t-2=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -3 for b, and -2 for c in the quadratic formula.

$$t=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 1\left(-2\right)}}{2}$$

Do the calculations.

$$t=\frac{3±\sqrt{17}}{2}$$

Solve the equation t=\frac{3±\sqrt{17}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{17}+3}{2} t=\frac{3-\sqrt{17}}{2}$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for positive t.

$$x=\frac{\sqrt{2\sqrt{17}+6}}{2} x=-\frac{\sqrt{2\sqrt{17}+6}}{2}$$