Solve x/2x^2+4x+x^2-5/2x

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

Evaluate

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

Short Solution Steps

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

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in x+x^{3}+2x^{2}-5x-10.

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

Expand 2x\left(x+2\right).

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

Expand

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

Short Solution Steps

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

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in x+x^{3}+2x^{2}-5x-10.

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

Expand 2x\left(x+2\right).

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