Solve n+4/4n+8+1/n^2+2n

$$\frac { n + 4 } { 4 n + 8 } + \frac { 1 } { n ^ { 2 } + 2 n }$$

Evaluate

$$\frac{1}{4}+\frac{1}{2n}$$

Short Solution Steps

Factor 4n+8. Factor n^{2}+2n.

$$\frac{n+4}{4\left(n+2\right)}+\frac{1}{n\left(n+2\right)}$$

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

$$\frac{\left(n+4\right)n}{4n\left(n+2\right)}+\frac{4}{4n\left(n+2\right)}$$

Since \frac{\left(n+4\right)n}{4n\left(n+2\right)} and \frac{4}{4n\left(n+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(n+4\right)n+4}{4n\left(n+2\right)}$$

Do the multiplications in \left(n+4\right)n+4.

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

Factor the expressions that are not already factored in \frac{n^{2}+4n+4}{4n\left(n+2\right)}.

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

Cancel out n+2 in both numerator and denominator.

$$\frac{n+2}{4n}$$

Expand

$$\frac{1}{4}+\frac{1}{2n}$$

Short Solution Steps

Factor 4n+8. Factor n^{2}+2n.

$$\frac{n+4}{4\left(n+2\right)}+\frac{1}{n\left(n+2\right)}$$

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

$$\frac{\left(n+4\right)n}{4n\left(n+2\right)}+\frac{4}{4n\left(n+2\right)}$$

Since \frac{\left(n+4\right)n}{4n\left(n+2\right)} and \frac{4}{4n\left(n+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(n+4\right)n+4}{4n\left(n+2\right)}$$

Do the multiplications in \left(n+4\right)n+4.

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

Factor the expressions that are not already factored in \frac{n^{2}+4n+4}{4n\left(n+2\right)}.

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

Cancel out n+2 in both numerator and denominator.

$$\frac{n+2}{4n}$$