Solve x/x+1-5/x+2=

$$\frac { x } { x + 1 } - \frac { 5 } { x + 2 } =$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and x+2 is \left(x+1\right)\left(x+2\right). Multiply \frac{x}{x+1} times \frac{x+2}{x+2}. Multiply \frac{5}{x+2} times \frac{x+1}{x+1}.

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

Since \frac{x\left(x+2\right)}{\left(x+1\right)\left(x+2\right)} and \frac{5\left(x+1\right)}{\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(x+2\right)-5\left(x+1\right).

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

Combine like terms in x^{2}+2x-5x-5.

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

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

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

Differentiate w.r.t. x

$$\frac{6x^{2}+14x+9}{\left(\left(x+1\right)\left(x+2\right)\right)^{2}}$$