Solve (x-2/x^2+2x-x-1/x^2+4x+4/frac{4-x{x}})

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

Evaluate

$$\frac{-2x^{3}+5x^{2}+4x-16}{x\left(4-x\right)\left(x+2\right)^{2}}$$

Short Solution Steps

Divide \frac{x-1}{x^{2}+4x+4} by \frac{4-x}{x} by multiplying \frac{x-1}{x^{2}+4x+4} by the reciprocal of \frac{4-x}{x}.

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

Factor x^{2}+2x. Factor \left(x^{2}+4x+4\right)\left(4-x\right).

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

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

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

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

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

Do the multiplications in \left(x-2\right)\left(x-4\right)\left(x+2\right)-\left(x-1\right)x\left(-1\right)x.

$$\frac{x^{3}-2x^{2}-8x-2x^{2}+4x+16+x^{3}-x^{2}}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Combine like terms in x^{3}-2x^{2}-8x-2x^{2}+4x+16+x^{3}-x^{2}.

$$\frac{2x^{3}-5x^{2}-4x+16}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Expand x\left(x-4\right)\left(x+2\right)^{2}.

$$\frac{2x^{3}-5x^{2}-4x+16}{x^{4}-12x^{2}-16x}$$

Expand

$$\frac{-2x^{3}+5x^{2}+4x-16}{x\left(4-x\right)\left(x+2\right)^{2}}$$

Short Solution Steps

Divide \frac{x-1}{x^{2}+4x+4} by \frac{4-x}{x} by multiplying \frac{x-1}{x^{2}+4x+4} by the reciprocal of \frac{4-x}{x}.

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

Factor x^{2}+2x. Factor \left(x^{2}+4x+4\right)\left(4-x\right).

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

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

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

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

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

Do the multiplications in \left(x-2\right)\left(x-4\right)\left(x+2\right)-\left(x-1\right)x\left(-1\right)x.

$$\frac{x^{3}-2x^{2}-8x-2x^{2}+4x+16+x^{3}-x^{2}}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Combine like terms in x^{3}-2x^{2}-8x-2x^{2}+4x+16+x^{3}-x^{2}.

$$\frac{2x^{3}-5x^{2}-4x+16}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Expand x\left(x-4\right)\left(x+2\right)^{2}.

$$\frac{2x^{3}-5x^{2}-4x+16}{x^{4}-12x^{2}-16x}$$