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

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

Evaluate

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

Short Solution Steps

Divide 1 by \frac{x-2}{x^{2}+2x+1} by multiplying 1 by the reciprocal of \frac{x-2}{x^{2}+2x+1}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x+1}{x+1}.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

Divide 1 by \frac{x-2}{x^{2}+2x+1} by multiplying 1 by the reciprocal of \frac{x-2}{x^{2}+2x+1}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x+1}{x+1}.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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