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

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

Short Solution Steps

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

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

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

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

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

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

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

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

Multiply \frac{\left(x-2\right)\left(x+2\right)^{2}}{\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.

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

Cancel out \left(x-2\right)\left(x+1\right)\left(x+2\right) in both numerator and denominator.

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

Use the distributive property to multiply x+1 by x+2 and combine like terms.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

Multiply \frac{\left(x-2\right)\left(x+2\right)^{2}}{\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.

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

Cancel out \left(x-2\right)\left(x+1\right)\left(x+2\right) in both numerator and denominator.

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

Use the distributive property to multiply x+1 by x+2 and combine like terms.

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