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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{\left(x-1\right)\left(x+3\right)}{\left(x-1\right)\left(x+3\right)}.

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

Since \frac{2x+2}{\left(x-1\right)\left(x+3\right)} and \frac{2\left(x-1\right)\left(x+3\right)}{\left(x-1\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2x+2-2\left(x-1\right)\left(x+3\right).

$$\frac{2x+2-2x^{2}-6x+2x+6}{\left(x-1\right)\left(x+3\right)}$$

Combine like terms in 2x+2-2x^{2}-6x+2x+6.

$$\frac{-2x+8-2x^{2}}{\left(x-1\right)\left(x+3\right)}$$

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

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

Differentiate w.r.t. x

$$-\frac{2\left(x^{2}+2x+5\right)}{\left(\left(x-1\right)\left(x+3\right)\right)^{2}}$$