Solve x+2/x+1/x+2/frac{5{x}+x/x+2}

$$\frac { \frac { x + 2 } { x } + \frac { 1 } { x + 2 } } { \frac { 5 } { x } + \frac { x } { x + 2 } }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

$$\frac{\frac{x^{2}+5x+4}{x\left(x+2\right)}}{\frac{5\left(x+2\right)}{x\left(x+2\right)}+\frac{xx}{x\left(x+2\right)}}$$

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

$$\frac{\frac{x^{2}+5x+4}{x\left(x+2\right)}}{\frac{5\left(x+2\right)+xx}{x\left(x+2\right)}}$$

Do the multiplications in 5\left(x+2\right)+xx.

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

Divide \frac{x^{2}+5x+4}{x\left(x+2\right)} by \frac{5x+10+x^{2}}{x\left(x+2\right)} by multiplying \frac{x^{2}+5x+4}{x\left(x+2\right)} by the reciprocal of \frac{5x+10+x^{2}}{x\left(x+2\right)}.

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

Cancel out x\left(x+2\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

$$\frac{\frac{x^{2}+5x+4}{x\left(x+2\right)}}{\frac{5\left(x+2\right)}{x\left(x+2\right)}+\frac{xx}{x\left(x+2\right)}}$$

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

$$\frac{\frac{x^{2}+5x+4}{x\left(x+2\right)}}{\frac{5\left(x+2\right)+xx}{x\left(x+2\right)}}$$

Do the multiplications in 5\left(x+2\right)+xx.

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

Divide \frac{x^{2}+5x+4}{x\left(x+2\right)} by \frac{5x+10+x^{2}}{x\left(x+2\right)} by multiplying \frac{x^{2}+5x+4}{x\left(x+2\right)} by the reciprocal of \frac{5x+10+x^{2}}{x\left(x+2\right)}.

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

Cancel out x\left(x+2\right) in both numerator and denominator.

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