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

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

Evaluate

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

Short Solution Steps

Combine -4x and x to get -3x.

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

Factor x^{2}-3x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x\left(x-3\right) is x\left(x-3\right)\left(x-2\right). Multiply \frac{x+1}{x-2} times \frac{x\left(x-3\right)}{x\left(x-3\right)}. Multiply \frac{2+x}{x\left(x-3\right)} times \frac{x-2}{x-2}.

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

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

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

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

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

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

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

Expand x\left(x-3\right)\left(x-2\right).

$$\frac{x^{3}-x^{2}-3x-4}{x^{3}-5x^{2}+6x}$$

Expand

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

Short Solution Steps

Combine -4x and x to get -3x.

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

Factor x^{2}-3x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x\left(x-3\right) is x\left(x-3\right)\left(x-2\right). Multiply \frac{x+1}{x-2} times \frac{x\left(x-3\right)}{x\left(x-3\right)}. Multiply \frac{2+x}{x\left(x-3\right)} times \frac{x-2}{x-2}.

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

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

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

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

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

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

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

Expand x\left(x-3\right)\left(x-2\right).

$$\frac{x^{3}-x^{2}-3x-4}{x^{3}-5x^{2}+6x}$$