Solve x+2/x^2+4x+3-5x/x^2-9

$$\frac { x + 2 } { x ^ { 2 } + 4 x + 3 } - \frac { 5 x } { x ^ { 2 } - 9 }$$

Evaluate

$$-\frac{2\left(2x^{2}+3x+3\right)}{\left(x+1\right)\left(x^{2}-9\right)}$$

Short Solution Steps

Factor x^{2}+4x+3. Factor x^{2}-9.

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

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-3\right)\left(x+3\right) is \left(x-3\right)\left(x+1\right)\left(x+3\right). Multiply \frac{x+2}{\left(x+1\right)\left(x+3\right)} times \frac{x-3}{x-3}. Multiply \frac{5x}{\left(x-3\right)\left(x+3\right)} times \frac{x+1}{x+1}.

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

Since \frac{\left(x+2\right)\left(x-3\right)}{\left(x-3\right)\left(x+1\right)\left(x+3\right)} and \frac{5x\left(x+1\right)}{\left(x-3\right)\left(x+1\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+2\right)\left(x-3\right)-5x\left(x+1\right).

$$\frac{x^{2}-3x+2x-6-5x^{2}-5x}{\left(x-3\right)\left(x+1\right)\left(x+3\right)}$$

Combine like terms in x^{2}-3x+2x-6-5x^{2}-5x.

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

Expand \left(x-3\right)\left(x+1\right)\left(x+3\right).

$$\frac{-4x^{2}-6x-6}{x^{3}+x^{2}-9x-9}$$

Expand

$$-\frac{2\left(2x^{2}+3x+3\right)}{\left(x+1\right)\left(x^{2}-9\right)}$$

Short Solution Steps

Factor x^{2}+4x+3. Factor x^{2}-9.

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

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-3\right)\left(x+3\right) is \left(x-3\right)\left(x+1\right)\left(x+3\right). Multiply \frac{x+2}{\left(x+1\right)\left(x+3\right)} times \frac{x-3}{x-3}. Multiply \frac{5x}{\left(x-3\right)\left(x+3\right)} times \frac{x+1}{x+1}.

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

Since \frac{\left(x+2\right)\left(x-3\right)}{\left(x-3\right)\left(x+1\right)\left(x+3\right)} and \frac{5x\left(x+1\right)}{\left(x-3\right)\left(x+1\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+2\right)\left(x-3\right)-5x\left(x+1\right).

$$\frac{x^{2}-3x+2x-6-5x^{2}-5x}{\left(x-3\right)\left(x+1\right)\left(x+3\right)}$$

Combine like terms in x^{2}-3x+2x-6-5x^{2}-5x.

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

Expand \left(x-3\right)\left(x+1\right)\left(x+3\right).

$$\frac{-4x^{2}-6x-6}{x^{3}+x^{2}-9x-9}$$