Solve f(x)=(x^2/1+x^2)*1-(-x^2/1+(-x^2))-2x

$$f ( x ) = ( \frac { x ^ { 2 } } { 1 + x ^ { 2 } } ) \cdot 1 - ( \frac { - x ^ { 2 } } { 1 + ( - x ^ { 2 } ) } ) - 2 x$$

Evaluate

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

Short Solution Steps

Express \frac{x^{2}}{1+x^{2}}\times 1 as a single fraction.

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

Factor 1-x^{2}.

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

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

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

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

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

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

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

Combine like terms in -x^{4}-x^{3}+x^{3}+x^{2}+x^{4}+x^{2}.

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

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

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

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

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

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

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

Combine like terms in 2x^{2}+2x^{5}+2x^{3}+2x^{4}+2x^{2}-2x^{4}-2x^{2}-2x^{3}-2x.

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

Expand \left(x-1\right)\left(-x-1\right)\left(x^{2}+1\right).

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

Factor

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