Solve frac{x+2-4x-1-8x^{-2}}{1+4x^-1+3x^-2}*x+2+x^-1/x-4x^-1

$$\frac { x + 2 - 4 x - 1 - 8 x ^ { - 2 } } { 1 + 4 x ^ { - 1 } + 3 x ^ { - 2 } } \cdot \frac { x + 2 + x ^ { - 1 } } { x - 4 x ^ { - 1 } }$$

Evaluate

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

Short Solution Steps

Combine x and -4x to get -3x.

$$\frac{-3x+2-1-8x^{-2}}{1+4x^{-1}+3x^{-2}}\times \frac{x+2+x^{-1}}{x-4x^{-1}}$$

Subtract 1 from 2 to get 1.

$$\frac{-3x+1-8x^{-2}}{1+4x^{-1}+3x^{-2}}\times \frac{x+2+x^{-1}}{x-4x^{-1}}$$

Factor the expressions that are not already factored in \frac{-3x+1-8x^{-2}}{1+4x^{-1}+3x^{-2}}.

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

Cancel out x^{-2} in both numerator and denominator.

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

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

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

Cancel out \frac{1}{x} in both numerator and denominator.

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

Multiply \frac{-\left(3x^{3}-x^{2}+8\right)}{\left(x+1\right)\left(x+3\right)} times \frac{\left(x+1\right)^{2}}{\left(x-2\right)\left(x+2\right)} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x+1 in both numerator and denominator.

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

Use the distributive property to multiply -1 by x+1.

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

Use the distributive property to multiply -x-1 by 3x^{3}-x^{2}+8 and combine like terms.

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

Use the distributive property to multiply x-2 by x+2 and combine like terms.

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

Use the distributive property to multiply x^{2}-4 by x+3.

$$\frac{-3x^{4}-2x^{3}-8x+x^{2}-8}{x^{3}+3x^{2}-4x-12}$$

Expand

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

Short Solution Steps

Combine x and -4x to get -3x.

$$\frac{-3x+2-1-8x^{-2}}{1+4x^{-1}+3x^{-2}}\times \frac{x+2+x^{-1}}{x-4x^{-1}}$$

Subtract 1 from 2 to get 1.

$$\frac{-3x+1-8x^{-2}}{1+4x^{-1}+3x^{-2}}\times \frac{x+2+x^{-1}}{x-4x^{-1}}$$

Factor the expressions that are not already factored in \frac{-3x+1-8x^{-2}}{1+4x^{-1}+3x^{-2}}.

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

Cancel out x^{-2} in both numerator and denominator.

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

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

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

Cancel out \frac{1}{x} in both numerator and denominator.

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

Multiply \frac{-\left(3x^{3}-x^{2}+8\right)}{\left(x+1\right)\left(x+3\right)} times \frac{\left(x+1\right)^{2}}{\left(x-2\right)\left(x+2\right)} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x+1 in both numerator and denominator.

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

Use the distributive property to multiply -1 by x+1.

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

Use the distributive property to multiply -x-1 by 3x^{3}-x^{2}+8 and combine like terms.

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

Use the distributive property to multiply x-2 by x+2 and combine like terms.

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

Use the distributive property to multiply x^{2}-4 by x+3.

$$\frac{-3x^{4}-2x^{3}-8x+x^{2}-8}{x^{3}+3x^{2}-4x-12}$$