Solve (x+1/x-2-1)x^2-2x/1x^2-4x+4/x^2+4

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-2}{x-2}.

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

Since \frac{x+1}{x-2} and \frac{x-2}{x-2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x+1-\left(x-2\right).

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

Combine like terms in x+1-x+2.

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

Anything divided by one gives itself.

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

Express \frac{3}{x-2}\left(x^{2}-2x\right) as a single fraction.

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

Multiply \frac{3\left(x^{2}-2x\right)}{x-2} times \frac{x^{2}-4x+4}{x^{2}+4} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-2 in both numerator and denominator.

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

Expand the expression.

$$\frac{3x^{3}-12x^{2}+12x}{x^{2}+4}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-2}{x-2}.

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

Since \frac{x+1}{x-2} and \frac{x-2}{x-2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x+1-\left(x-2\right).

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

Combine like terms in x+1-x+2.

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

Anything divided by one gives itself.

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

Express \frac{3}{x-2}\left(x^{2}-2x\right) as a single fraction.

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

Multiply \frac{3\left(x^{2}-2x\right)}{x-2} times \frac{x^{2}-4x+4}{x^{2}+4} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-2 in both numerator and denominator.

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

Expand the expression.

$$\frac{3x^{3}-12x^{2}+12x}{x^{2}+4}$$