Solve 2x^2-2/x^2-2x-8+x^2-1/x^2+5x+4

$$\frac { 2 x ^ { 2 } - 2 } { x ^ { 2 } - 2 x - 8 } + \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 5 x + 4 }$$

Evaluate

$$\frac{x\left(x-1\right)\left(3x+8\right)}{\left(x+2\right)\left(x^{2}-16\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-1}{x^{2}+5x+4}.

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

Cancel out x+1 in both numerator and denominator.

$$\frac{2x^{2}-2}{x^{2}-2x-8}+\frac{x-1}{x+4}$$

Factor x^{2}-2x-8.

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

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

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

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

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

Do the multiplications in \left(2x^{2}-2\right)\left(x+4\right)+\left(x-1\right)\left(x-4\right)\left(x+2\right).

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

Combine like terms in 2x^{3}+8x^{2}-2x-8+x^{3}-2x^{2}-8x-x^{2}+2x+8.

$$\frac{3x^{3}+5x^{2}-8x}{\left(x-4\right)\left(x+2\right)\left(x+4\right)}$$

Expand \left(x-4\right)\left(x+2\right)\left(x+4\right).

$$\frac{3x^{3}+5x^{2}-8x}{x^{3}+2x^{2}-16x-32}$$

Expand

$$\frac{3x^{3}+5x^{2}-8x}{\left(x+4\right)\left(x^{2}-2x-8\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-1}{x^{2}+5x+4}.

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

Cancel out x+1 in both numerator and denominator.

$$\frac{2x^{2}-2}{x^{2}-2x-8}+\frac{x-1}{x+4}$$

Factor x^{2}-2x-8.

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

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

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

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

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

Do the multiplications in \left(2x^{2}-2\right)\left(x+4\right)+\left(x-1\right)\left(x-4\right)\left(x+2\right).

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

Combine like terms in 2x^{3}+8x^{2}-2x-8+x^{3}-2x^{2}-8x-x^{2}+2x+8.

$$\frac{3x^{3}+5x^{2}-8x}{\left(x-4\right)\left(x+2\right)\left(x+4\right)}$$

Expand \left(x-4\right)\left(x+2\right)\left(x+4\right).

$$\frac{3x^{3}+5x^{2}-8x}{x^{3}+2x^{2}-16x-32}$$