Solve frac{x^{2}}{2-x}divx^2+5x/x^2-4*x^2-25/x^2-2x-10

$$\frac { x ^ { 2 } } { 2 - x } \div \frac { x ^ { 2 } + 5 x } { x ^ { 2 } - 4 } \times \frac { x ^ { 2 } - 25 } { x ^ { 2 } - 2 x - 10 }$$

Evaluate

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

Short Solution Steps

Divide \frac{x^{2}}{2-x} by \frac{x^{2}+5x}{x^{2}-4} by multiplying \frac{x^{2}}{2-x} by the reciprocal of \frac{x^{2}+5x}{x^{2}-4}.

$$\frac{x^{2}\left(x^{2}-4\right)}{\left(2-x\right)\left(x^{2}+5x\right)}\times \frac{x^{2}-25}{x^{2}-2x-10}$$

Factor the expressions that are not already factored in \frac{x^{2}\left(x^{2}-4\right)}{\left(2-x\right)\left(x^{2}+5x\right)}.

$$\frac{\left(x-2\right)\left(x+2\right)x^{2}}{x\left(x+5\right)\left(-x+2\right)}\times \frac{x^{2}-25}{x^{2}-2x-10}$$

Extract the negative sign in -2+x.

$$\frac{-\left(x+2\right)\left(-x+2\right)x^{2}}{x\left(x+5\right)\left(-x+2\right)}\times \frac{x^{2}-25}{x^{2}-2x-10}$$

Cancel out x\left(-x+2\right) in both numerator and denominator.

$$\frac{-x\left(x+2\right)}{x+5}\times \frac{x^{2}-25}{x^{2}-2x-10}$$

Multiply \frac{-x\left(x+2\right)}{x+5} times \frac{x^{2}-25}{x^{2}-2x-10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-x\left(x+2\right)\left(x^{2}-25\right)}{\left(x+5\right)\left(x^{2}-2x-10\right)}$$

Factor the expressions that are not already factored.

$$\frac{-x\left(x-5\right)\left(x+2\right)\left(x+5\right)}{\left(x+5\right)\left(x-\left(\sqrt{11}+1\right)\right)\left(x-\left(-\sqrt{11}+1\right)\right)}$$

Cancel out x+5 in both numerator and denominator.

$$\frac{-x\left(x-5\right)\left(x+2\right)}{\left(x-\left(\sqrt{11}+1\right)\right)\left(x-\left(-\sqrt{11}+1\right)\right)}$$

Expand the expression.

$$\frac{-x^{3}+3x^{2}+10x}{x^{2}-2x-10}$$

Expand

$$-\frac{x^{3}-3x^{2}-10x}{x^{2}-2x-10}$$

Short Solution Steps

Divide \frac{x^{2}}{2-x} by \frac{x^{2}+5x}{x^{2}-4} by multiplying \frac{x^{2}}{2-x} by the reciprocal of \frac{x^{2}+5x}{x^{2}-4}.

$$\frac{x^{2}\left(x^{2}-4\right)}{\left(2-x\right)\left(x^{2}+5x\right)}\times \frac{x^{2}-25}{x^{2}-2x-10}$$

Factor the expressions that are not already factored in \frac{x^{2}\left(x^{2}-4\right)}{\left(2-x\right)\left(x^{2}+5x\right)}.

$$\frac{\left(x-2\right)\left(x+2\right)x^{2}}{x\left(x+5\right)\left(-x+2\right)}\times \frac{x^{2}-25}{x^{2}-2x-10}$$

Extract the negative sign in -2+x.

$$\frac{-\left(x+2\right)\left(-x+2\right)x^{2}}{x\left(x+5\right)\left(-x+2\right)}\times \frac{x^{2}-25}{x^{2}-2x-10}$$

Cancel out x\left(-x+2\right) in both numerator and denominator.

$$\frac{-x\left(x+2\right)}{x+5}\times \frac{x^{2}-25}{x^{2}-2x-10}$$

Multiply \frac{-x\left(x+2\right)}{x+5} times \frac{x^{2}-25}{x^{2}-2x-10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-x\left(x+2\right)\left(x^{2}-25\right)}{\left(x+5\right)\left(x^{2}-2x-10\right)}$$

Factor the expressions that are not already factored.

$$\frac{-x\left(x-5\right)\left(x+2\right)\left(x+5\right)}{\left(x+5\right)\left(x-\left(\sqrt{11}+1\right)\right)\left(x-\left(-\sqrt{11}+1\right)\right)}$$

Cancel out x+5 in both numerator and denominator.

$$\frac{-x\left(x-5\right)\left(x+2\right)}{\left(x-\left(\sqrt{11}+1\right)\right)\left(x-\left(-\sqrt{11}+1\right)\right)}$$

Expand the expression.

$$\frac{-x^{3}+3x^{2}+10x}{x^{2}-2x-10}$$