Solve 1/x^2-2x-3+1/x^2-9

$$\frac { 1 } { x ^ { 2 } - 2 x - 3 } + \frac { 1 } { x ^ { 2 } - 9 }$$

Evaluate

$$\frac{2\left(x+2\right)}{\left(x+1\right)\left(x^{2}-9\right)}$$

Short Solution Steps

Factor x^{2}-2x-3. Factor x^{2}-9.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+1\right) and \left(x-3\right)\left(x+3\right) is \left(x-3\right)\left(x+1\right)\left(x+3\right). Multiply \frac{1}{\left(x-3\right)\left(x+1\right)} times \frac{x+3}{x+3}. Multiply \frac{1}{\left(x-3\right)\left(x+3\right)} times \frac{x+1}{x+1}.

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

Since \frac{x+3}{\left(x-3\right)\left(x+1\right)\left(x+3\right)} and \frac{x+1}{\left(x-3\right)\left(x+1\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in x+3+x+1.

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

Expand \left(x-3\right)\left(x+1\right)\left(x+3\right).

$$\frac{2x+4}{x^{3}+x^{2}-9x-9}$$

Differentiate w.r.t. x

$$\frac{2\left(9-4x-7x^{2}-2x^{3}\right)}{\left(\left(x+1\right)\left(x^{2}-9\right)\right)^{2}}$$