Solve x-1/x^2+4x+3+x/x^2-9-x+3/x^2-2x-3

$$\frac { x - 1 } { x ^ { 2 } + 4 x + 3 } + \frac { x } { x ^ { 2 } - 9 } - \frac { x + 3 } { x ^ { 2 } - 2 x - 3 }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Do the multiplications in 2x^{2}-3x+3-\left(x+3\right)\left(x+3\right).

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

Combine like terms in 2x^{2}-3x+3-x^{2}-3x-3x-9.

$$\frac{x^{2}-9x-6}{\left(x-3\right)\left(x+1\right)\left(x+3\right)}$$

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

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Do the multiplications in 2x^{2}-3x+3-\left(x+3\right)\left(x+3\right).

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

Combine like terms in 2x^{2}-3x+3-x^{2}-3x-3x-9.

$$\frac{x^{2}-9x-6}{\left(x-3\right)\left(x+1\right)\left(x+3\right)}$$

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

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