Solve x^2+x+1/x^2+x-12+x+2/x+4-x-2/x-3

$$\frac { x ^ { 2 } + x + 1 } { x ^ { 2 } + x - 12 } + \frac { x + 2 } { x + 4 } - \frac { x - 2 } { x - 3 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}+x-12.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right) and x+4 is \left(x-3\right)\left(x+4\right). Multiply \frac{x+2}{x+4} times \frac{x-3}{x-3}.

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

Since \frac{x^{2}+x+1}{\left(x-3\right)\left(x+4\right)} and \frac{\left(x+2\right)\left(x-3\right)}{\left(x-3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

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

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

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

Combine like terms in x^{2}+x+1+x^{2}-3x+2x-6.

$$\frac{2x^{2}-5}{\left(x-3\right)\left(x+4\right)}-\frac{x-2}{x-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right) and x-3 is \left(x-3\right)\left(x+4\right). Multiply \frac{x-2}{x-3} times \frac{x+4}{x+4}.

$$\frac{2x^{2}-5}{\left(x-3\right)\left(x+4\right)}-\frac{\left(x-2\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)}$$

Since \frac{2x^{2}-5}{\left(x-3\right)\left(x+4\right)} and \frac{\left(x-2\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x^{2}-5-\left(x-2\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)}$$

Do the multiplications in 2x^{2}-5-\left(x-2\right)\left(x+4\right).

$$\frac{2x^{2}-5-x^{2}-4x+2x+8}{\left(x-3\right)\left(x+4\right)}$$

Combine like terms in 2x^{2}-5-x^{2}-4x+2x+8.

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

Expand \left(x-3\right)\left(x+4\right).

$$\frac{x^{2}+3-2x}{x^{2}+x-12}$$

Expand

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

Short Solution Steps

Factor x^{2}+x-12.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right) and x+4 is \left(x-3\right)\left(x+4\right). Multiply \frac{x+2}{x+4} times \frac{x-3}{x-3}.

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

Since \frac{x^{2}+x+1}{\left(x-3\right)\left(x+4\right)} and \frac{\left(x+2\right)\left(x-3\right)}{\left(x-3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

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

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

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

Combine like terms in x^{2}+x+1+x^{2}-3x+2x-6.

$$\frac{2x^{2}-5}{\left(x-3\right)\left(x+4\right)}-\frac{x-2}{x-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right) and x-3 is \left(x-3\right)\left(x+4\right). Multiply \frac{x-2}{x-3} times \frac{x+4}{x+4}.

$$\frac{2x^{2}-5}{\left(x-3\right)\left(x+4\right)}-\frac{\left(x-2\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)}$$

Since \frac{2x^{2}-5}{\left(x-3\right)\left(x+4\right)} and \frac{\left(x-2\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x^{2}-5-\left(x-2\right)\left(x+4\right)}{\left(x-3\right)\left(x+4\right)}$$

Do the multiplications in 2x^{2}-5-\left(x-2\right)\left(x+4\right).

$$\frac{2x^{2}-5-x^{2}-4x+2x+8}{\left(x-3\right)\left(x+4\right)}$$

Combine like terms in 2x^{2}-5-x^{2}-4x+2x+8.

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

Expand \left(x-3\right)\left(x+4\right).

$$\frac{x^{2}+3-2x}{x^{2}+x-12}$$