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

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

Evaluate

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

Short Solution Steps

Factor x^{2}+3x+2.

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored in \frac{-x^{2}-2x}{\left(x+1\right)\left(x+2\right)}.

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

Extract the negative sign in -2-x.

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

Cancel out x+2 in both numerator and denominator.

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

Factor

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