Solve x^2-2x/x^2-4-x^4+x-2/x^2+4x+4

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

Evaluate

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

Short Solution Steps

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

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

Cancel out x-2 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

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

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

Cancel out x-2 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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