Solve x-2/x^2-16+x-4/2-x

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

Evaluate

$$\frac{\left(x-5\right)\left(x^{2}-12\right)}{\left(2-x\right)\left(x^{2}-16\right)}$$

Short Solution Steps

Factor x^{2}-16.

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

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

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

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

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

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

$$\frac{-x^{2}+2x+2x-4+x^{3}-16x-4x^{2}+64}{\left(x-4\right)\left(x+4\right)\left(-x+2\right)}$$

Combine like terms in -x^{2}+2x+2x-4+x^{3}-16x-4x^{2}+64.

$$\frac{-5x^{2}-12x+60+x^{3}}{\left(x-4\right)\left(x+4\right)\left(-x+2\right)}$$

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

$$\frac{-5x^{2}-12x+60+x^{3}}{-x^{3}+2x^{2}+16x-32}$$

Expand

$$\frac{x^{3}-5x^{2}-12x+60}{\left(2-x\right)\left(x^{2}-16\right)}$$

Short Solution Steps

Factor x^{2}-16.

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

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

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

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

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

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

$$\frac{-x^{2}+2x+2x-4+x^{3}-16x-4x^{2}+64}{\left(x-4\right)\left(x+4\right)\left(-x+2\right)}$$

Combine like terms in -x^{2}+2x+2x-4+x^{3}-16x-4x^{2}+64.

$$\frac{-5x^{2}-12x+60+x^{3}}{\left(x-4\right)\left(x+4\right)\left(-x+2\right)}$$

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

$$\frac{-5x^{2}-12x+60+x^{3}}{-x^{3}+2x^{2}+16x-32}$$