Solve x^2+3x+2x/2x(x/x^2-1-x/x-5)+5/5+x

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

Evaluate

$$\frac{50-110x-65x^{2}-9x^{3}-9x^{4}-x^{5}}{2\left(x^{2}-25\right)\left(x^{2}-1\right)}$$

Short Solution Steps

Combine 3x and 2x to get 5x.

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

Factor the expressions that are not already factored in \frac{x^{2}+5x}{2x}.

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

Cancel out x in both numerator and denominator.

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

Factor x^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and x-5 is \left(x-5\right)\left(x-1\right)\left(x+1\right). Multiply \frac{x}{\left(x-1\right)\left(x+1\right)} times \frac{x-5}{x-5}. Multiply \frac{x}{x-5} times \frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}.

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

Since \frac{x\left(x-5\right)}{\left(x-5\right)\left(x-1\right)\left(x+1\right)} and \frac{x\left(x-1\right)\left(x+1\right)}{\left(x-5\right)\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(x-5\right)-x\left(x-1\right)\left(x+1\right).

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

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

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

Multiply \frac{x+5}{2} times \frac{x^{2}-4x-x^{3}}{\left(x-5\right)\left(x-1\right)\left(x+1\right)} by multiplying numerator times numerator and denominator times denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-5\right)\left(x-1\right)\left(x+1\right) and 5+x is 2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right). Multiply \frac{\left(x+5\right)\left(x^{2}-4x-x^{3}\right)}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)} times \frac{x+5}{x+5}. Multiply \frac{5}{5+x} times \frac{2\left(x-5\right)\left(x-1\right)\left(x+1\right)}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)}.

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

Since \frac{\left(x+5\right)\left(x^{2}-4x-x^{3}\right)\left(x+5\right)}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right)} and \frac{5\times 2\left(x-5\right)\left(x-1\right)\left(x+1\right)}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x+5\right)\left(x^{2}-4x-x^{3}\right)\left(x+5\right)+5\times 2\left(x-5\right)\left(x-1\right)\left(x+1\right).

$$\frac{-4x^{4}+x^{3}-20x^{2}-x^{5}-20x^{3}+5x^{2}-100x-5x^{4}+10x^{3}-10x-50x^{2}+50}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right)}$$

Combine like terms in -4x^{4}+x^{3}-20x^{2}-x^{5}-20x^{3}+5x^{2}-100x-5x^{4}+10x^{3}-10x-50x^{2}+50.

$$\frac{-9x^{4}-9x^{3}-65x^{2}-x^{5}-110x+50}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right)}$$

Expand 2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right).

$$\frac{-9x^{4}-9x^{3}-65x^{2}-x^{5}-110x+50}{2x^{4}-52x^{2}+50}$$

Expand

$$-\frac{x^{5}+9x^{4}+9x^{3}+65x^{2}+110x-50}{2\left(x^{2}-25\right)\left(x^{2}-1\right)}$$

Short Solution Steps

Combine 3x and 2x to get 5x.

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

Factor the expressions that are not already factored in \frac{x^{2}+5x}{2x}.

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

Cancel out x in both numerator and denominator.

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

Factor x^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and x-5 is \left(x-5\right)\left(x-1\right)\left(x+1\right). Multiply \frac{x}{\left(x-1\right)\left(x+1\right)} times \frac{x-5}{x-5}. Multiply \frac{x}{x-5} times \frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}.

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

Since \frac{x\left(x-5\right)}{\left(x-5\right)\left(x-1\right)\left(x+1\right)} and \frac{x\left(x-1\right)\left(x+1\right)}{\left(x-5\right)\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(x-5\right)-x\left(x-1\right)\left(x+1\right).

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

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

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

Multiply \frac{x+5}{2} times \frac{x^{2}-4x-x^{3}}{\left(x-5\right)\left(x-1\right)\left(x+1\right)} by multiplying numerator times numerator and denominator times denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-5\right)\left(x-1\right)\left(x+1\right) and 5+x is 2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right). Multiply \frac{\left(x+5\right)\left(x^{2}-4x-x^{3}\right)}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)} times \frac{x+5}{x+5}. Multiply \frac{5}{5+x} times \frac{2\left(x-5\right)\left(x-1\right)\left(x+1\right)}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)}.

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

Since \frac{\left(x+5\right)\left(x^{2}-4x-x^{3}\right)\left(x+5\right)}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right)} and \frac{5\times 2\left(x-5\right)\left(x-1\right)\left(x+1\right)}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x+5\right)\left(x^{2}-4x-x^{3}\right)\left(x+5\right)+5\times 2\left(x-5\right)\left(x-1\right)\left(x+1\right).

$$\frac{-4x^{4}+x^{3}-20x^{2}-x^{5}-20x^{3}+5x^{2}-100x-5x^{4}+10x^{3}-10x-50x^{2}+50}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right)}$$

Combine like terms in -4x^{4}+x^{3}-20x^{2}-x^{5}-20x^{3}+5x^{2}-100x-5x^{4}+10x^{3}-10x-50x^{2}+50.

$$\frac{-9x^{4}-9x^{3}-65x^{2}-x^{5}-110x+50}{2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right)}$$

Expand 2\left(x-5\right)\left(x-1\right)\left(x+1\right)\left(x+5\right).

$$\frac{-9x^{4}-9x^{3}-65x^{2}-x^{5}-110x+50}{2x^{4}-52x^{2}+50}$$