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

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

Evaluate

$$\frac{x^{2}-3x+16}{\left(x+1\right)\left(x^{2}-16\right)}$$

Short Solution Steps

Factor x^{2}-16. Factor x^{2}+5x+4.

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

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 \left(x+1\right)\left(x+4\right) is \left(x-4\right)\left(x+1\right)\left(x+4\right). Multiply \frac{x}{\left(x-4\right)\left(x+4\right)} times \frac{x+1}{x+1}. Multiply \frac{4}{\left(x+1\right)\left(x+4\right)} times \frac{x-4}{x-4}.

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

Since \frac{x\left(x+1\right)}{\left(x-4\right)\left(x+1\right)\left(x+4\right)} and \frac{4\left(x-4\right)}{\left(x-4\right)\left(x+1\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(x+1\right)-4\left(x-4\right).

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

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

$$\frac{x^{2}-3x+16}{\left(x-4\right)\left(x+1\right)\left(x+4\right)}$$

Expand \left(x-4\right)\left(x+1\right)\left(x+4\right).

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

Differentiate w.r.t. x

$$\frac{304-64x-61x^{2}+6x^{3}-x^{4}}{\left(\left(x+1\right)\left(x^{2}-16\right)\right)^{2}}$$