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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

Expand \left(x-1\right)\left(x+2\right).

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

Expand \left(x-1\right)\left(x+2\right).

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