Solve x/x^2-3x-4-2x/x^2-1+x^2-6x-4/x^3-4{x^2-x+4}

$$\frac{ x }{ { x }^{ 2 } -3x-4 } - \frac{ 2x }{ { x }^{ 2 } -1 } + \frac{ { x }^{ 2 } -6x-4 }{ { x }^{ 3 } -4 { x }^{ 2 } -x+4 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-3x-4. Factor x^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4\right)\left(x+1\right) and \left(x-1\right)\left(x+1\right) is \left(x-4\right)\left(x-1\right)\left(x+1\right). Multiply \frac{x}{\left(x-4\right)\left(x+1\right)} times \frac{x-1}{x-1}. Multiply \frac{2x}{\left(x-1\right)\left(x+1\right)} times \frac{x-4}{x-4}.

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

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

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

Do the multiplications in x\left(x-1\right)-2x\left(x-4\right).

$$\frac{x^{2}-x-2x^{2}+8x}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}+\frac{x^{2}-6x-4}{x^{3}-4x^{2}-x+4}$$

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

$$\frac{-x^{2}+7x}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}+\frac{x^{2}-6x-4}{x^{3}-4x^{2}-x+4}$$

Factor x^{3}-4x^{2}-x+4.

$$\frac{-x^{2}+7x}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}+\frac{x^{2}-6x-4}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Since \frac{-x^{2}+7x}{\left(x-4\right)\left(x-1\right)\left(x+1\right)} and \frac{x^{2}-6x-4}{\left(x-4\right)\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{-x^{2}+7x+x^{2}-6x-4}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Combine like terms in -x^{2}+7x+x^{2}-6x-4.

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

Cancel out x-4 in both numerator and denominator.

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

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

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

Expand

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

Short Solution Steps

Factor x^{2}-3x-4. Factor x^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4\right)\left(x+1\right) and \left(x-1\right)\left(x+1\right) is \left(x-4\right)\left(x-1\right)\left(x+1\right). Multiply \frac{x}{\left(x-4\right)\left(x+1\right)} times \frac{x-1}{x-1}. Multiply \frac{2x}{\left(x-1\right)\left(x+1\right)} times \frac{x-4}{x-4}.

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

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

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

Do the multiplications in x\left(x-1\right)-2x\left(x-4\right).

$$\frac{x^{2}-x-2x^{2}+8x}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}+\frac{x^{2}-6x-4}{x^{3}-4x^{2}-x+4}$$

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

$$\frac{-x^{2}+7x}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}+\frac{x^{2}-6x-4}{x^{3}-4x^{2}-x+4}$$

Factor x^{3}-4x^{2}-x+4.

$$\frac{-x^{2}+7x}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}+\frac{x^{2}-6x-4}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Since \frac{-x^{2}+7x}{\left(x-4\right)\left(x-1\right)\left(x+1\right)} and \frac{x^{2}-6x-4}{\left(x-4\right)\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{-x^{2}+7x+x^{2}-6x-4}{\left(x-4\right)\left(x-1\right)\left(x+1\right)}$$

Combine like terms in -x^{2}+7x+x^{2}-6x-4.

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

Cancel out x-4 in both numerator and denominator.

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

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

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