Solve 3/x-4+8/x+4+3+x^2/x^2+16

$$\frac { 3 } { x - 4 } + \frac { 8 } { x + 4 } + \frac { 3 + x ^ { 2 } } { x ^ { 2 } + 16 }$$

Evaluate

$$\frac{x^{4}+11x^{3}-33x^{2}+176x-368}{x^{4}-256}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-4 and x+4 is \left(x-4\right)\left(x+4\right). Multiply \frac{3}{x-4} times \frac{x+4}{x+4}. Multiply \frac{8}{x+4} times \frac{x-4}{x-4}.

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

Since \frac{3\left(x+4\right)}{\left(x-4\right)\left(x+4\right)} and \frac{8\left(x-4\right)}{\left(x-4\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\left(x+4\right)+8\left(x-4\right).

$$\frac{3x+12+8x-32}{\left(x-4\right)\left(x+4\right)}+\frac{3+x^{2}}{x^{2}+16}$$

Combine like terms in 3x+12+8x-32.

$$\frac{11x-20}{\left(x-4\right)\left(x+4\right)}+\frac{3+x^{2}}{x^{2}+16}$$

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 x^{2}+16 is \left(x-4\right)\left(x+4\right)\left(x^{2}+16\right). Multiply \frac{11x-20}{\left(x-4\right)\left(x+4\right)} times \frac{x^{2}+16}{x^{2}+16}. Multiply \frac{3+x^{2}}{x^{2}+16} times \frac{\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+4\right)}.

$$\frac{\left(11x-20\right)\left(x^{2}+16\right)}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)}+\frac{\left(3+x^{2}\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)}$$

Since \frac{\left(11x-20\right)\left(x^{2}+16\right)}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)} and \frac{\left(3+x^{2}\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(11x-20\right)\left(x^{2}+16\right)+\left(3+x^{2}\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)}$$

Do the multiplications in \left(11x-20\right)\left(x^{2}+16\right)+\left(3+x^{2}\right)\left(x-4\right)\left(x+4\right).

$$\frac{11x^{3}+176x-20x^{2}-320+3x^{2}-48+x^{4}-16x^{2}}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)}$$

Combine like terms in 11x^{3}+176x-20x^{2}-320+3x^{2}-48+x^{4}-16x^{2}.

$$\frac{11x^{3}+176x-33x^{2}-368+x^{4}}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)}$$

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

$$\frac{11x^{3}+176x-33x^{2}-368+x^{4}}{x^{4}-256}$$

Expand

$$\frac{x^{4}+11x^{3}-33x^{2}+176x-368}{x^{4}-256}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-4 and x+4 is \left(x-4\right)\left(x+4\right). Multiply \frac{3}{x-4} times \frac{x+4}{x+4}. Multiply \frac{8}{x+4} times \frac{x-4}{x-4}.

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

Since \frac{3\left(x+4\right)}{\left(x-4\right)\left(x+4\right)} and \frac{8\left(x-4\right)}{\left(x-4\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\left(x+4\right)+8\left(x-4\right).

$$\frac{3x+12+8x-32}{\left(x-4\right)\left(x+4\right)}+\frac{3+x^{2}}{x^{2}+16}$$

Combine like terms in 3x+12+8x-32.

$$\frac{11x-20}{\left(x-4\right)\left(x+4\right)}+\frac{3+x^{2}}{x^{2}+16}$$

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 x^{2}+16 is \left(x-4\right)\left(x+4\right)\left(x^{2}+16\right). Multiply \frac{11x-20}{\left(x-4\right)\left(x+4\right)} times \frac{x^{2}+16}{x^{2}+16}. Multiply \frac{3+x^{2}}{x^{2}+16} times \frac{\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+4\right)}.

$$\frac{\left(11x-20\right)\left(x^{2}+16\right)}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)}+\frac{\left(3+x^{2}\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)}$$

Since \frac{\left(11x-20\right)\left(x^{2}+16\right)}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)} and \frac{\left(3+x^{2}\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(11x-20\right)\left(x^{2}+16\right)+\left(3+x^{2}\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)}$$

Do the multiplications in \left(11x-20\right)\left(x^{2}+16\right)+\left(3+x^{2}\right)\left(x-4\right)\left(x+4\right).

$$\frac{11x^{3}+176x-20x^{2}-320+3x^{2}-48+x^{4}-16x^{2}}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)}$$

Combine like terms in 11x^{3}+176x-20x^{2}-320+3x^{2}-48+x^{4}-16x^{2}.

$$\frac{11x^{3}+176x-33x^{2}-368+x^{4}}{\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)}$$

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

$$\frac{11x^{3}+176x-33x^{2}-368+x^{4}}{x^{4}-256}$$