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

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

Evaluate

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

Short Solution Steps

Factor x^{2}+3x+2. Factor 1x^{2}-1.

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

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 \left(x-1\right)\left(x+1\right) is \left(x-1\right)\left(x+1\right)\left(x+2\right). Multiply \frac{x}{\left(x+1\right)\left(x+2\right)} times \frac{x-1}{x-1}. Multiply \frac{2x-3}{\left(x-1\right)\left(x+1\right)} times \frac{x+2}{x+2}.

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

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

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

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

$$\frac{x^{2}-x+2x^{2}+4x-3x-6}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Combine like terms in x^{2}-x+2x^{2}+4x-3x-6.

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

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

$$\frac{3x^{2}-6}{x^{3}+2x^{2}-x-2}$$

Expand

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

Short Solution Steps

Factor x^{2}+3x+2. Factor 1x^{2}-1.

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

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 \left(x-1\right)\left(x+1\right) is \left(x-1\right)\left(x+1\right)\left(x+2\right). Multiply \frac{x}{\left(x+1\right)\left(x+2\right)} times \frac{x-1}{x-1}. Multiply \frac{2x-3}{\left(x-1\right)\left(x+1\right)} times \frac{x+2}{x+2}.

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

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

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

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

$$\frac{x^{2}-x+2x^{2}+4x-3x-6}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Combine like terms in x^{2}-x+2x^{2}+4x-3x-6.

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

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

$$\frac{3x^{2}-6}{x^{3}+2x^{2}-x-2}$$