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

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

Evaluate

$$\frac{2\left(x^{7}-4x^{5}+8x^{4}+16x^{3}+32x^{2}+128\right)}{\left(x^{4}+16\right)^{2}-16x^{4}}$$

Short Solution Steps

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right) and x^{4}-4x^{2}+16 is \left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right). Multiply \frac{2x^{3}}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)} times \frac{x^{4}-4x^{2}+16}{x^{4}-4x^{2}+16}. Multiply \frac{16}{x^{4}-4x^{2}+16} times \frac{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)}.

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

Since \frac{2x^{3}\left(x^{4}-4x^{2}+16\right)}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right)} and \frac{16\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2x^{3}\left(x^{4}-4x^{2}+16\right)+16\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right).

$$\frac{2x^{7}-8x^{5}+32x^{3}+16x^{4}+32x^{3}+64x^{2}-32x^{3}-64x^{2}-128x+64x^{2}+128x+256}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right)}$$

Combine like terms in 2x^{7}-8x^{5}+32x^{3}+16x^{4}+32x^{3}+64x^{2}-32x^{3}-64x^{2}-128x+64x^{2}+128x+256.

$$\frac{2x^{7}-8x^{5}+32x^{3}+16x^{4}+64x^{2}+256}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right)}$$

Expand \left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right).

$$\frac{2x^{7}-8x^{5}+32x^{3}+16x^{4}+64x^{2}+256}{x^{8}+16x^{4}+256}$$

Expand

$$\frac{2\left(x^{7}-4x^{5}+8x^{4}+16x^{3}+32x^{2}+128\right)}{\left(x^{4}+16\right)^{2}-16x^{4}}$$

Short Solution Steps

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right) and x^{4}-4x^{2}+16 is \left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right). Multiply \frac{2x^{3}}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)} times \frac{x^{4}-4x^{2}+16}{x^{4}-4x^{2}+16}. Multiply \frac{16}{x^{4}-4x^{2}+16} times \frac{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)}.

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

Since \frac{2x^{3}\left(x^{4}-4x^{2}+16\right)}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right)} and \frac{16\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2x^{3}\left(x^{4}-4x^{2}+16\right)+16\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right).

$$\frac{2x^{7}-8x^{5}+32x^{3}+16x^{4}+32x^{3}+64x^{2}-32x^{3}-64x^{2}-128x+64x^{2}+128x+256}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right)}$$

Combine like terms in 2x^{7}-8x^{5}+32x^{3}+16x^{4}+32x^{3}+64x^{2}-32x^{3}-64x^{2}-128x+64x^{2}+128x+256.

$$\frac{2x^{7}-8x^{5}+32x^{3}+16x^{4}+64x^{2}+256}{\left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right)}$$

Expand \left(x^{2}-2x+4\right)\left(x^{2}+2x+4\right)\left(x^{4}-4x^{2}+16\right).

$$\frac{2x^{7}-8x^{5}+32x^{3}+16x^{4}+64x^{2}+256}{x^{8}+16x^{4}+256}$$