Solve x+2/2-x/x+2+2x^2-x^3/2x^2-8

$$\frac { x + 2 } { 2 } - \frac { x } { x + 2 } + \frac { 2 x ^ { 2 } - x ^ { 3 } } { 2 x ^ { 2 } - 8 }$$

Evaluate

$$1$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{2x^{2}-x^{3}}{2x^{2}-8}.

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

Extract the negative sign in 2-x.

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

Cancel out x-2 in both numerator and denominator.

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

Cancel out -1 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored in \frac{2x+4}{2\left(x+2\right)}.

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

Cancel out 2\left(x+2\right) in both numerator and denominator.

$$1$$

Factor

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