Solve 1+2x/1+x^2/2x+frac{2x^2+2{1-x^4}}=

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Extract the negative sign in x^{2}+1.

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

Cancel out -x^{2}-1 in both numerator and denominator.

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

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-1\right)\left(x+1\right)}.

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

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

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

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

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

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

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

Divide \frac{1+x^{2}+2x}{1+x^{2}} by \frac{2x^{3}-2x-2}{\left(x-1\right)\left(x+1\right)} by multiplying \frac{1+x^{2}+2x}{1+x^{2}} by the reciprocal of \frac{2x^{3}-2x-2}{\left(x-1\right)\left(x+1\right)}.

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

Extract the negative sign in x^{2}+1.

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

Cancel out -x^{2}-1 in both numerator and denominator.

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

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-1\right)\left(x+1\right)}.

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

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

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

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

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

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

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

Divide \frac{1+x^{2}+2x}{1+x^{2}} by \frac{2x^{3}-2x-2}{\left(x-1\right)\left(x+1\right)} by multiplying \frac{1+x^{2}+2x}{1+x^{2}} by the reciprocal of \frac{2x^{3}-2x-2}{\left(x-1\right)\left(x+1\right)}.

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

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

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

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

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

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

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