Solve x+1/x+2+2x-3/2-x+x^2+2x+4/x^2-4

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

Evaluate

$$\frac{8}{x^{2}-4}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and 2-x is \left(x+2\right)\left(-x+2\right). Multiply \frac{x+1}{x+2} times \frac{-x+2}{-x+2}. Multiply \frac{2x-3}{2-x} times \frac{x+2}{x+2}.

$$\frac{\left(x+1\right)\left(-x+2\right)}{\left(x+2\right)\left(-x+2\right)}+\frac{\left(2x-3\right)\left(x+2\right)}{\left(x+2\right)\left(-x+2\right)}+\frac{x^{2}+2x+4}{x^{2}-4}$$

Since \frac{\left(x+1\right)\left(-x+2\right)}{\left(x+2\right)\left(-x+2\right)} and \frac{\left(2x-3\right)\left(x+2\right)}{\left(x+2\right)\left(-x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+1\right)\left(-x+2\right)+\left(2x-3\right)\left(x+2\right)}{\left(x+2\right)\left(-x+2\right)}+\frac{x^{2}+2x+4}{x^{2}-4}$$

Do the multiplications in \left(x+1\right)\left(-x+2\right)+\left(2x-3\right)\left(x+2\right).

$$\frac{-x^{2}+2x+2-x+2x^{2}+4x-3x-6}{\left(x+2\right)\left(-x+2\right)}+\frac{x^{2}+2x+4}{x^{2}-4}$$

Combine like terms in -x^{2}+2x+2-x+2x^{2}+4x-3x-6.

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

Factor x^{2}-4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)\left(-x+2\right) and \left(x-2\right)\left(x+2\right) is \left(x-2\right)\left(x+2\right). Multiply \frac{x^{2}+2x-4}{\left(x+2\right)\left(-x+2\right)} times \frac{-1}{-1}.

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

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

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

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

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

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

$$\frac{8}{\left(x-2\right)\left(x+2\right)}$$

Expand \left(x-2\right)\left(x+2\right).

$$\frac{8}{x^{2}-4}$$

Expand

$$\frac{8}{x^{2}-4}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and 2-x is \left(x+2\right)\left(-x+2\right). Multiply \frac{x+1}{x+2} times \frac{-x+2}{-x+2}. Multiply \frac{2x-3}{2-x} times \frac{x+2}{x+2}.

$$\frac{\left(x+1\right)\left(-x+2\right)}{\left(x+2\right)\left(-x+2\right)}+\frac{\left(2x-3\right)\left(x+2\right)}{\left(x+2\right)\left(-x+2\right)}+\frac{x^{2}+2x+4}{x^{2}-4}$$

Since \frac{\left(x+1\right)\left(-x+2\right)}{\left(x+2\right)\left(-x+2\right)} and \frac{\left(2x-3\right)\left(x+2\right)}{\left(x+2\right)\left(-x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+1\right)\left(-x+2\right)+\left(2x-3\right)\left(x+2\right)}{\left(x+2\right)\left(-x+2\right)}+\frac{x^{2}+2x+4}{x^{2}-4}$$

Do the multiplications in \left(x+1\right)\left(-x+2\right)+\left(2x-3\right)\left(x+2\right).

$$\frac{-x^{2}+2x+2-x+2x^{2}+4x-3x-6}{\left(x+2\right)\left(-x+2\right)}+\frac{x^{2}+2x+4}{x^{2}-4}$$

Combine like terms in -x^{2}+2x+2-x+2x^{2}+4x-3x-6.

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

Factor x^{2}-4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)\left(-x+2\right) and \left(x-2\right)\left(x+2\right) is \left(x-2\right)\left(x+2\right). Multiply \frac{x^{2}+2x-4}{\left(x+2\right)\left(-x+2\right)} times \frac{-1}{-1}.

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

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

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

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

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

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

$$\frac{8}{\left(x-2\right)\left(x+2\right)}$$

Expand \left(x-2\right)\left(x+2\right).

$$\frac{8}{x^{2}-4}$$