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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x-1+x^{2}+2x.

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

Expand \left(x-1\right)\left(x+2\right).

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

Differentiate w.r.t. x

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