Solve (1+x)(1-x)((1+x)^5-(1-x)^5)/2x(1+x)^6

$$\frac { ( 1 + x ) ( 1 - x ) ( ( 1 + x ) ^ { 5 } - ( 1 - x ) ^ { 5 } ) } { 2 x ( 1 + x ) ^ { 6 } }$$

Evaluate

$$\frac{\left(1-x\right)\left(x^{4}+10x^{2}+5\right)}{\left(x+1\right)^{5}}$$

Short Solution Steps

Cancel out x+1 in both numerator and denominator.

$$\frac{\left(-x+1\right)\left(\left(x+1\right)^{5}-\left(-x+1\right)^{5}\right)}{2x\left(x+1\right)^{5}}$$

Factor the expressions that are not already factored.

$$\frac{2x\left(-x+1\right)\left(x^{2}-\left(-2\sqrt{5}-5\right)\right)\left(x^{2}-\left(2\sqrt{5}-5\right)\right)}{2x\left(x+1\right)^{5}}$$

Cancel out 2x in both numerator and denominator.

$$\frac{\left(-x+1\right)\left(x^{2}-\left(-2\sqrt{5}-5\right)\right)\left(x^{2}-\left(2\sqrt{5}-5\right)\right)}{\left(x+1\right)^{5}}$$

Expand the expression.

$$\frac{-x^{5}+x^{4}-10x^{3}+10x^{2}-5x+5}{x^{5}+5x^{4}+10x^{3}+10x^{2}+5x+1}$$

Expand

$$\frac{5-5x+10x^{2}-10x^{3}+x^{4}-x^{5}}{\left(x+1\right)^{5}}$$

Short Solution Steps

Cancel out x+1 in both numerator and denominator.

$$\frac{\left(-x+1\right)\left(\left(x+1\right)^{5}-\left(-x+1\right)^{5}\right)}{2x\left(x+1\right)^{5}}$$

Factor the expressions that are not already factored.

$$\frac{2x\left(-x+1\right)\left(x^{2}-\left(-2\sqrt{5}-5\right)\right)\left(x^{2}-\left(2\sqrt{5}-5\right)\right)}{2x\left(x+1\right)^{5}}$$

Cancel out 2x in both numerator and denominator.

$$\frac{\left(-x+1\right)\left(x^{2}-\left(-2\sqrt{5}-5\right)\right)\left(x^{2}-\left(2\sqrt{5}-5\right)\right)}{\left(x+1\right)^{5}}$$

Expand the expression.

$$\frac{-x^{5}+x^{4}-10x^{3}+10x^{2}-5x+5}{x^{5}+5x^{4}+10x^{3}+10x^{2}+5x+1}$$