Solve 5/x+3-4/x^2+3x+16/x^2-4

$$\frac { 5 } { x + 3 } - \frac { 4 } { x ^ { 2 } + 3 x } + \frac { 16 } { x ^ { 2 } - 4 }$$

Evaluate

$$\frac{5x^{3}+12x^{2}+28x+16}{x\left(x+3\right)\left(x^{2}-4\right)}$$

Short Solution Steps

Factor x^{2}+3x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+3 and x\left(x+3\right) is x\left(x+3\right). Multiply \frac{5}{x+3} times \frac{x}{x}.

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

Since \frac{5x}{x\left(x+3\right)} and \frac{4}{x\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Factor x^{2}-4.

$$\frac{5x-4}{x\left(x+3\right)}+\frac{16}{\left(x-2\right)\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+3\right) and \left(x-2\right)\left(x+2\right) is x\left(x-2\right)\left(x+2\right)\left(x+3\right). Multiply \frac{5x-4}{x\left(x+3\right)} times \frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}. Multiply \frac{16}{\left(x-2\right)\left(x+2\right)} times \frac{x\left(x+3\right)}{x\left(x+3\right)}.

$$\frac{\left(5x-4\right)\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)\left(x+3\right)}+\frac{16x\left(x+3\right)}{x\left(x-2\right)\left(x+2\right)\left(x+3\right)}$$

Since \frac{\left(5x-4\right)\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)\left(x+3\right)} and \frac{16x\left(x+3\right)}{x\left(x-2\right)\left(x+2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(5x-4\right)\left(x-2\right)\left(x+2\right)+16x\left(x+3\right)}{x\left(x-2\right)\left(x+2\right)\left(x+3\right)}$$

Do the multiplications in \left(5x-4\right)\left(x-2\right)\left(x+2\right)+16x\left(x+3\right).

$$\frac{5x^{3}-20x-4x^{2}+16+16x^{2}+48x}{x\left(x-2\right)\left(x+2\right)\left(x+3\right)}$$

Combine like terms in 5x^{3}-20x-4x^{2}+16+16x^{2}+48x.

$$\frac{5x^{3}+28x+12x^{2}+16}{x\left(x-2\right)\left(x+2\right)\left(x+3\right)}$$

Expand x\left(x-2\right)\left(x+2\right)\left(x+3\right).

$$\frac{5x^{3}+28x+12x^{2}+16}{x^{4}+3x^{3}-4x^{2}-12x}$$

Differentiate w.r.t. x

$$\frac{192+128x-176x^{2}-352x^{3}-140x^{4}-24x^{5}-5x^{6}}{\left(x\left(x+3\right)\left(x^{2}-4\right)\right)^{2}}$$