Solve 5-x/x^2+2x-3+3x+5/x^2+4x+3

$$\frac { 5 - x } { x ^ { 2 } + 2 x - 3 } + \frac { 3 x + 5 } { x ^ { 2 } + 4 x + 3 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}+2x-3. Factor x^{2}+4x+3.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+3\right) and \left(x+1\right)\left(x+3\right) is \left(x-1\right)\left(x+1\right)\left(x+3\right). Multiply \frac{5-x}{\left(x-1\right)\left(x+3\right)} times \frac{x+1}{x+1}. Multiply \frac{3x+5}{\left(x+1\right)\left(x+3\right)} times \frac{x-1}{x-1}.

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

Since \frac{\left(5-x\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x+3\right)} and \frac{\left(3x+5\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(5-x\right)\left(x+1\right)+\left(3x+5\right)\left(x-1\right).

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

Combine like terms in 5x+5-x^{2}-x+3x^{2}-3x+5x-5.

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

Factor the expressions that are not already factored in \frac{6x+2x^{2}}{\left(x-1\right)\left(x+1\right)\left(x+3\right)}.

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

Cancel out x+3 in both numerator and denominator.

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

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

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

Expand

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

Short Solution Steps

Factor x^{2}+2x-3. Factor x^{2}+4x+3.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+3\right) and \left(x+1\right)\left(x+3\right) is \left(x-1\right)\left(x+1\right)\left(x+3\right). Multiply \frac{5-x}{\left(x-1\right)\left(x+3\right)} times \frac{x+1}{x+1}. Multiply \frac{3x+5}{\left(x+1\right)\left(x+3\right)} times \frac{x-1}{x-1}.

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

Since \frac{\left(5-x\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x+3\right)} and \frac{\left(3x+5\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(5-x\right)\left(x+1\right)+\left(3x+5\right)\left(x-1\right).

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

Combine like terms in 5x+5-x^{2}-x+3x^{2}-3x+5x-5.

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

Factor the expressions that are not already factored in \frac{6x+2x^{2}}{\left(x-1\right)\left(x+1\right)\left(x+3\right)}.

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

Cancel out x+3 in both numerator and denominator.

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

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

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