Solve frac{x^{2}-2x-8}{x^{3}-4x^2-4x+16}+x^2+4x-32/x^2-16+24/x^2+2x-8

$$\frac { x ^ { 2 } - 2 x - 8 } { x ^ { 3 } - 4 x ^ { 2 } - 4 x + 16 } + \frac { x ^ { 2 } + 4 x - 32 } { x ^ { 2 } - 16 } + \frac { 24 } { x ^ { 2 } + 2 x - 8 }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-2x-8}{x^{3}-4x^{2}-4x+16}.

$$\frac{\left(x-4\right)\left(x+2\right)}{\left(x-4\right)\left(x-2\right)\left(x+2\right)}+\frac{x^{2}+4x-32}{x^{2}-16}+\frac{24}{x^{2}+2x-8}$$

Cancel out \left(x-4\right)\left(x+2\right) in both numerator and denominator.

$$\frac{1}{x-2}+\frac{x^{2}+4x-32}{x^{2}-16}+\frac{24}{x^{2}+2x-8}$$

Factor the expressions that are not already factored in \frac{x^{2}+4x-32}{x^{2}-16}.

$$\frac{1}{x-2}+\frac{\left(x-4\right)\left(x+8\right)}{\left(x-4\right)\left(x+4\right)}+\frac{24}{x^{2}+2x-8}$$

Cancel out x-4 in both numerator and denominator.

$$\frac{1}{x-2}+\frac{x+8}{x+4}+\frac{24}{x^{2}+2x-8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x+4 is \left(x-2\right)\left(x+4\right). Multiply \frac{1}{x-2} times \frac{x+4}{x+4}. Multiply \frac{x+8}{x+4} times \frac{x-2}{x-2}.

$$\frac{x+4}{\left(x-2\right)\left(x+4\right)}+\frac{\left(x+8\right)\left(x-2\right)}{\left(x-2\right)\left(x+4\right)}+\frac{24}{x^{2}+2x-8}$$

Since \frac{x+4}{\left(x-2\right)\left(x+4\right)} and \frac{\left(x+8\right)\left(x-2\right)}{\left(x-2\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+4+\left(x+8\right)\left(x-2\right)}{\left(x-2\right)\left(x+4\right)}+\frac{24}{x^{2}+2x-8}$$

Do the multiplications in x+4+\left(x+8\right)\left(x-2\right).

$$\frac{x+4+x^{2}-2x+8x-16}{\left(x-2\right)\left(x+4\right)}+\frac{24}{x^{2}+2x-8}$$

Combine like terms in x+4+x^{2}-2x+8x-16.

$$\frac{7x-12+x^{2}}{\left(x-2\right)\left(x+4\right)}+\frac{24}{x^{2}+2x-8}$$

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

$$\frac{7x-12+x^{2}}{\left(x-2\right)\left(x+4\right)}+\frac{24}{\left(x-2\right)\left(x+4\right)}$$

Since \frac{7x-12+x^{2}}{\left(x-2\right)\left(x+4\right)} and \frac{24}{\left(x-2\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{7x-12+x^{2}+24}{\left(x-2\right)\left(x+4\right)}$$

Combine like terms in 7x-12+x^{2}+24.

$$\frac{7x+12+x^{2}}{\left(x-2\right)\left(x+4\right)}$$

Factor the expressions that are not already factored in \frac{7x+12+x^{2}}{\left(x-2\right)\left(x+4\right)}.

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

Cancel out x+4 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-2x-8}{x^{3}-4x^{2}-4x+16}.

$$\frac{\left(x-4\right)\left(x+2\right)}{\left(x-4\right)\left(x-2\right)\left(x+2\right)}+\frac{x^{2}+4x-32}{x^{2}-16}+\frac{24}{x^{2}+2x-8}$$

Cancel out \left(x-4\right)\left(x+2\right) in both numerator and denominator.

$$\frac{1}{x-2}+\frac{x^{2}+4x-32}{x^{2}-16}+\frac{24}{x^{2}+2x-8}$$

Factor the expressions that are not already factored in \frac{x^{2}+4x-32}{x^{2}-16}.

$$\frac{1}{x-2}+\frac{\left(x-4\right)\left(x+8\right)}{\left(x-4\right)\left(x+4\right)}+\frac{24}{x^{2}+2x-8}$$

Cancel out x-4 in both numerator and denominator.

$$\frac{1}{x-2}+\frac{x+8}{x+4}+\frac{24}{x^{2}+2x-8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x+4 is \left(x-2\right)\left(x+4\right). Multiply \frac{1}{x-2} times \frac{x+4}{x+4}. Multiply \frac{x+8}{x+4} times \frac{x-2}{x-2}.

$$\frac{x+4}{\left(x-2\right)\left(x+4\right)}+\frac{\left(x+8\right)\left(x-2\right)}{\left(x-2\right)\left(x+4\right)}+\frac{24}{x^{2}+2x-8}$$

Since \frac{x+4}{\left(x-2\right)\left(x+4\right)} and \frac{\left(x+8\right)\left(x-2\right)}{\left(x-2\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+4+\left(x+8\right)\left(x-2\right)}{\left(x-2\right)\left(x+4\right)}+\frac{24}{x^{2}+2x-8}$$

Do the multiplications in x+4+\left(x+8\right)\left(x-2\right).

$$\frac{x+4+x^{2}-2x+8x-16}{\left(x-2\right)\left(x+4\right)}+\frac{24}{x^{2}+2x-8}$$

Combine like terms in x+4+x^{2}-2x+8x-16.

$$\frac{7x-12+x^{2}}{\left(x-2\right)\left(x+4\right)}+\frac{24}{x^{2}+2x-8}$$

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

$$\frac{7x-12+x^{2}}{\left(x-2\right)\left(x+4\right)}+\frac{24}{\left(x-2\right)\left(x+4\right)}$$

Since \frac{7x-12+x^{2}}{\left(x-2\right)\left(x+4\right)} and \frac{24}{\left(x-2\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{7x-12+x^{2}+24}{\left(x-2\right)\left(x+4\right)}$$

Combine like terms in 7x-12+x^{2}+24.

$$\frac{7x+12+x^{2}}{\left(x-2\right)\left(x+4\right)}$$

Factor the expressions that are not already factored in \frac{7x+12+x^{2}}{\left(x-2\right)\left(x+4\right)}.

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

Cancel out x+4 in both numerator and denominator.

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