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

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

Evaluate

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

Short Solution Steps

Factor x^{2}+2x. Factor x^{2}+4x+4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right) and \left(x+2\right)^{2} is x\left(x+2\right)^{2}. Multiply \frac{x-2}{x\left(x+2\right)} times \frac{x+2}{x+2}. Multiply \frac{x-1}{\left(x+2\right)^{2}} times \frac{x}{x}.

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right)^{2} and x is x\left(x+2\right)^{2}. Multiply \frac{4-x}{x} times \frac{\left(x+2\right)^{2}}{\left(x+2\right)^{2}}.

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

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

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

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

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

Combine like terms in x-4-4x^{2}-16x-16+x^{3}+4x^{2}+4x.

$$\frac{-11x-20+x^{3}}{x\left(x+2\right)^{2}}$$

Expand x\left(x+2\right)^{2}.

$$\frac{-11x-20+x^{3}}{x^{3}+4x^{2}+4x}$$

Expand

$$\frac{x^{3}-11x-20}{x\left(x+2\right)^{2}}$$

Short Solution Steps

Factor x^{2}+2x. Factor x^{2}+4x+4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right) and \left(x+2\right)^{2} is x\left(x+2\right)^{2}. Multiply \frac{x-2}{x\left(x+2\right)} times \frac{x+2}{x+2}. Multiply \frac{x-1}{\left(x+2\right)^{2}} times \frac{x}{x}.

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right)^{2} and x is x\left(x+2\right)^{2}. Multiply \frac{4-x}{x} times \frac{\left(x+2\right)^{2}}{\left(x+2\right)^{2}}.

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

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

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

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

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

Combine like terms in x-4-4x^{2}-16x-16+x^{3}+4x^{2}+4x.

$$\frac{-11x-20+x^{3}}{x\left(x+2\right)^{2}}$$

Expand x\left(x+2\right)^{2}.

$$\frac{-11x-20+x^{3}}{x^{3}+4x^{2}+4x}$$