Solve 1/x-1+-7/x+1+10/left(x+1right)^2

$$\frac{ 1 }{ x-1 } + \frac{ -7 }{ x+1 } + \frac{ 10 }{ { \left(x+1 \right) }^{ 2 } }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x+1 is \left(x-1\right)\left(x+1\right). Multiply \frac{1}{x-1} times \frac{x+1}{x+1}. Multiply \frac{-7}{x+1} times \frac{x-1}{x-1}.

$$\frac{x+1}{\left(x-1\right)\left(x+1\right)}+\frac{-7\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{10}{\left(x+1\right)^{2}}$$

Since \frac{x+1}{\left(x-1\right)\left(x+1\right)} and \frac{-7\left(x-1\right)}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+1-7\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{10}{\left(x+1\right)^{2}}$$

Do the multiplications in x+1-7\left(x-1\right).

$$\frac{x+1-7x+7}{\left(x-1\right)\left(x+1\right)}+\frac{10}{\left(x+1\right)^{2}}$$

Combine like terms in x+1-7x+7.

$$\frac{-6x+8}{\left(x-1\right)\left(x+1\right)}+\frac{10}{\left(x+1\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and \left(x+1\right)^{2} is \left(x-1\right)\left(x+1\right)^{2}. Multiply \frac{-6x+8}{\left(x-1\right)\left(x+1\right)} times \frac{x+1}{x+1}. Multiply \frac{10}{\left(x+1\right)^{2}} times \frac{x-1}{x-1}.

$$\frac{\left(-6x+8\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)^{2}}+\frac{10\left(x-1\right)}{\left(x-1\right)\left(x+1\right)^{2}}$$

Since \frac{\left(-6x+8\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)^{2}} and \frac{10\left(x-1\right)}{\left(x-1\right)\left(x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-6x+8\right)\left(x+1\right)+10\left(x-1\right)}{\left(x-1\right)\left(x+1\right)^{2}}$$

Do the multiplications in \left(-6x+8\right)\left(x+1\right)+10\left(x-1\right).

$$\frac{-6x^{2}-6x+8x+8+10x-10}{\left(x-1\right)\left(x+1\right)^{2}}$$

Combine like terms in -6x^{2}-6x+8x+8+10x-10.

$$\frac{-6x^{2}+12x-2}{\left(x-1\right)\left(x+1\right)^{2}}$$

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

$$\frac{-6x^{2}+12x-2}{x^{3}+x^{2}-x-1}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x+1 is \left(x-1\right)\left(x+1\right). Multiply \frac{1}{x-1} times \frac{x+1}{x+1}. Multiply \frac{-7}{x+1} times \frac{x-1}{x-1}.

$$\frac{x+1}{\left(x-1\right)\left(x+1\right)}+\frac{-7\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{10}{\left(x+1\right)^{2}}$$

Since \frac{x+1}{\left(x-1\right)\left(x+1\right)} and \frac{-7\left(x-1\right)}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+1-7\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{10}{\left(x+1\right)^{2}}$$

Do the multiplications in x+1-7\left(x-1\right).

$$\frac{x+1-7x+7}{\left(x-1\right)\left(x+1\right)}+\frac{10}{\left(x+1\right)^{2}}$$

Combine like terms in x+1-7x+7.

$$\frac{-6x+8}{\left(x-1\right)\left(x+1\right)}+\frac{10}{\left(x+1\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and \left(x+1\right)^{2} is \left(x-1\right)\left(x+1\right)^{2}. Multiply \frac{-6x+8}{\left(x-1\right)\left(x+1\right)} times \frac{x+1}{x+1}. Multiply \frac{10}{\left(x+1\right)^{2}} times \frac{x-1}{x-1}.

$$\frac{\left(-6x+8\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)^{2}}+\frac{10\left(x-1\right)}{\left(x-1\right)\left(x+1\right)^{2}}$$

Since \frac{\left(-6x+8\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)^{2}} and \frac{10\left(x-1\right)}{\left(x-1\right)\left(x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-6x+8\right)\left(x+1\right)+10\left(x-1\right)}{\left(x-1\right)\left(x+1\right)^{2}}$$

Do the multiplications in \left(-6x+8\right)\left(x+1\right)+10\left(x-1\right).

$$\frac{-6x^{2}-6x+8x+8+10x-10}{\left(x-1\right)\left(x+1\right)^{2}}$$

Combine like terms in -6x^{2}-6x+8x+8+10x-10.

$$\frac{-6x^{2}+12x-2}{\left(x-1\right)\left(x+1\right)^{2}}$$

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

$$\frac{-6x^{2}+12x-2}{x^{3}+x^{2}-x-1}$$