Solve x^2-2x/x^2-4x*x^2-5x+6/x^2+x-2

$$\frac { x ^ { 2 } - 2 x } { x ^ { 2 } - 4 x } \cdot \frac { x ^ { 2 } - 5 x + 6 } { x ^ { 2 } + x - 2 }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-2x}{x^{2}-4x}.

$$\frac{x\left(x-2\right)}{x\left(x-4\right)}\times \frac{x^{2}-5x+6}{x^{2}+x-2}$$

Cancel out x in both numerator and denominator.

$$\frac{x-2}{x-4}\times \frac{x^{2}-5x+6}{x^{2}+x-2}$$

Multiply \frac{x-2}{x-4} times \frac{x^{2}-5x+6}{x^{2}+x-2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x-2\right)\left(x^{2}-5x+6\right)}{\left(x-4\right)\left(x^{2}+x-2\right)}$$

Use the distributive property to multiply x-2 by x^{2}-5x+6 and combine like terms.

$$\frac{x^{3}-7x^{2}+16x-12}{\left(x-4\right)\left(x^{2}+x-2\right)}$$

Use the distributive property to multiply x-4 by x^{2}+x-2 and combine like terms.

$$\frac{x^{3}-7x^{2}+16x-12}{x^{3}-3x^{2}-6x+8}$$

Expand

$$\frac{x^{3}-7x^{2}+16x-12}{\left(x-4\right)\left(x^{2}+x-2\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-2x}{x^{2}-4x}.

$$\frac{x\left(x-2\right)}{x\left(x-4\right)}\times \frac{x^{2}-5x+6}{x^{2}+x-2}$$

Cancel out x in both numerator and denominator.

$$\frac{x-2}{x-4}\times \frac{x^{2}-5x+6}{x^{2}+x-2}$$

Multiply \frac{x-2}{x-4} times \frac{x^{2}-5x+6}{x^{2}+x-2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x-2\right)\left(x^{2}-5x+6\right)}{\left(x-4\right)\left(x^{2}+x-2\right)}$$

Use the distributive property to multiply x-2 by x^{2}-5x+6 and combine like terms.

$$\frac{x^{3}-7x^{2}+16x-12}{\left(x-4\right)\left(x^{2}+x-2\right)}$$

Use the distributive property to multiply x-4 by x^{2}+x-2 and combine like terms.

$$\frac{x^{3}-7x^{2}+16x-12}{x^{3}-3x^{2}-6x+8}$$