Solve x/1x+16-x^2+16/(2x-8)(2x+8)-4/x^2-4x=

$$\frac { x } { 1 x + 16 } - \frac { x ^ { 2 } + 16 } { ( 2 x - 8 ) ( 2 x + 8 ) } - \frac { 4 } { x ^ { 2 } - 4 x } =$$

Evaluate

$$\frac{3x^{4}-16x^{3}-96x^{2}-576x-1024}{4x\left(x+16\right)\left(x^{2}-16\right)}$$

Short Solution Steps

Factor \left(2x-8\right)\left(2x+8\right).

$$\frac{x}{1x+16}-\frac{x^{2}+16}{2^{2}\left(x-4\right)\left(x+4\right)}-\frac{4}{x^{2}-4x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1x+16 and 2^{2}\left(x-4\right)\left(x+4\right) is 2^{2}\left(x-4\right)\left(x+4\right)\left(x+16\right). Multiply \frac{x}{1x+16} times \frac{2^{2}\left(x-4\right)\left(x+4\right)}{2^{2}\left(x-4\right)\left(x+4\right)}. Multiply \frac{x^{2}+16}{2^{2}\left(x-4\right)\left(x+4\right)} times \frac{x+16}{x+16}.

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

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

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

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

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

Combine like terms in 4x^{3}+16x^{2}-16x^{2}-64x-x^{3}-16x^{2}-16x-256.

$$\frac{3x^{3}-16x^{2}-80x-256}{2^{2}\left(x-4\right)\left(x+4\right)\left(x+16\right)}-\frac{4}{x^{2}-4x}$$

Factor x^{2}-4x.

$$\frac{3x^{3}-16x^{2}-80x-256}{2^{2}\left(x-4\right)\left(x+4\right)\left(x+16\right)}-\frac{4}{x\left(x-4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2^{2}\left(x-4\right)\left(x+4\right)\left(x+16\right) and x\left(x-4\right) is 2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right). Multiply \frac{3x^{3}-16x^{2}-80x-256}{2^{2}\left(x-4\right)\left(x+4\right)\left(x+16\right)} times \frac{x}{x}. Multiply \frac{4}{x\left(x-4\right)} times \frac{2^{2}\left(x+4\right)\left(x+16\right)}{2^{2}\left(x+4\right)\left(x+16\right)}.

$$\frac{\left(3x^{3}-16x^{2}-80x-256\right)x}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)}-\frac{4\times 2^{2}\left(x+4\right)\left(x+16\right)}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)}$$

Since \frac{\left(3x^{3}-16x^{2}-80x-256\right)x}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)} and \frac{4\times 2^{2}\left(x+4\right)\left(x+16\right)}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3x^{3}-16x^{2}-80x-256\right)x-4\times 2^{2}\left(x+4\right)\left(x+16\right)}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)}$$

Do the multiplications in \left(3x^{3}-16x^{2}-80x-256\right)x-4\times 2^{2}\left(x+4\right)\left(x+16\right).

$$\frac{3x^{4}-16x^{3}-80x^{2}-256x-16x^{2}-256x-64x-1024}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)}$$

Combine like terms in 3x^{4}-16x^{3}-80x^{2}-256x-16x^{2}-256x-64x-1024.

$$\frac{3x^{4}-16x^{3}-96x^{2}-576x-1024}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)}$$

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

$$\frac{3x^{4}-16x^{3}-96x^{2}-576x-1024}{4x^{4}+64x^{3}-64x^{2}-1024x}$$

Expand

$$\frac{3x^{4}-16x^{3}-96x^{2}-576x-1024}{4\left(x+4\right)\left(x+16\right)\left(x^{2}-4x\right)}$$

Short Solution Steps

Factor \left(2x-8\right)\left(2x+8\right).

$$\frac{x}{1x+16}-\frac{x^{2}+16}{2^{2}\left(x-4\right)\left(x+4\right)}-\frac{4}{x^{2}-4x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1x+16 and 2^{2}\left(x-4\right)\left(x+4\right) is 2^{2}\left(x-4\right)\left(x+4\right)\left(x+16\right). Multiply \frac{x}{1x+16} times \frac{2^{2}\left(x-4\right)\left(x+4\right)}{2^{2}\left(x-4\right)\left(x+4\right)}. Multiply \frac{x^{2}+16}{2^{2}\left(x-4\right)\left(x+4\right)} times \frac{x+16}{x+16}.

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

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

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

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

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

Combine like terms in 4x^{3}+16x^{2}-16x^{2}-64x-x^{3}-16x^{2}-16x-256.

$$\frac{3x^{3}-16x^{2}-80x-256}{2^{2}\left(x-4\right)\left(x+4\right)\left(x+16\right)}-\frac{4}{x^{2}-4x}$$

Factor x^{2}-4x.

$$\frac{3x^{3}-16x^{2}-80x-256}{2^{2}\left(x-4\right)\left(x+4\right)\left(x+16\right)}-\frac{4}{x\left(x-4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2^{2}\left(x-4\right)\left(x+4\right)\left(x+16\right) and x\left(x-4\right) is 2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right). Multiply \frac{3x^{3}-16x^{2}-80x-256}{2^{2}\left(x-4\right)\left(x+4\right)\left(x+16\right)} times \frac{x}{x}. Multiply \frac{4}{x\left(x-4\right)} times \frac{2^{2}\left(x+4\right)\left(x+16\right)}{2^{2}\left(x+4\right)\left(x+16\right)}.

$$\frac{\left(3x^{3}-16x^{2}-80x-256\right)x}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)}-\frac{4\times 2^{2}\left(x+4\right)\left(x+16\right)}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)}$$

Since \frac{\left(3x^{3}-16x^{2}-80x-256\right)x}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)} and \frac{4\times 2^{2}\left(x+4\right)\left(x+16\right)}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3x^{3}-16x^{2}-80x-256\right)x-4\times 2^{2}\left(x+4\right)\left(x+16\right)}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)}$$

Do the multiplications in \left(3x^{3}-16x^{2}-80x-256\right)x-4\times 2^{2}\left(x+4\right)\left(x+16\right).

$$\frac{3x^{4}-16x^{3}-80x^{2}-256x-16x^{2}-256x-64x-1024}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)}$$

Combine like terms in 3x^{4}-16x^{3}-80x^{2}-256x-16x^{2}-256x-64x-1024.

$$\frac{3x^{4}-16x^{3}-96x^{2}-576x-1024}{2^{2}x\left(x-4\right)\left(x+4\right)\left(x+16\right)}$$

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

$$\frac{3x^{4}-16x^{3}-96x^{2}-576x-1024}{4x^{4}+64x^{3}-64x^{2}-1024x}$$