Solve u+2/u^2+4/u^2+2u/frac{6{u^2}+u^2/u+2}

$$\frac { \frac { u + 2 } { u ^ { 2 } } + \frac { 4 } { u ^ { 2 } + 2 u } } { \frac { 6 } { u ^ { 2 } } + \frac { u ^ { 2 } } { u + 2 } }$$

Evaluate

$$\frac{u^{2}+8u+4}{u^{4}+6u+12}$$

Short Solution Steps

Factor u^{2}+2u.

$$\frac{\frac{u+2}{u^{2}}+\frac{4}{u\left(u+2\right)}}{\frac{6}{u^{2}}+\frac{u^{2}}{u+2}}$$

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

$$\frac{\frac{\left(u+2\right)\left(u+2\right)}{\left(u+2\right)u^{2}}+\frac{4u}{\left(u+2\right)u^{2}}}{\frac{6}{u^{2}}+\frac{u^{2}}{u+2}}$$

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

$$\frac{\frac{\left(u+2\right)\left(u+2\right)+4u}{\left(u+2\right)u^{2}}}{\frac{6}{u^{2}}+\frac{u^{2}}{u+2}}$$

Do the multiplications in \left(u+2\right)\left(u+2\right)+4u.

$$\frac{\frac{u^{2}+2u+2u+4+4u}{\left(u+2\right)u^{2}}}{\frac{6}{u^{2}}+\frac{u^{2}}{u+2}}$$

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

$$\frac{\frac{u^{2}+8u+4}{\left(u+2\right)u^{2}}}{\frac{6}{u^{2}}+\frac{u^{2}}{u+2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of u^{2} and u+2 is \left(u+2\right)u^{2}. Multiply \frac{6}{u^{2}} times \frac{u+2}{u+2}. Multiply \frac{u^{2}}{u+2} times \frac{u^{2}}{u^{2}}.

$$\frac{\frac{u^{2}+8u+4}{\left(u+2\right)u^{2}}}{\frac{6\left(u+2\right)}{\left(u+2\right)u^{2}}+\frac{u^{2}u^{2}}{\left(u+2\right)u^{2}}}$$

Since \frac{6\left(u+2\right)}{\left(u+2\right)u^{2}} and \frac{u^{2}u^{2}}{\left(u+2\right)u^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{u^{2}+8u+4}{\left(u+2\right)u^{2}}}{\frac{6\left(u+2\right)+u^{2}u^{2}}{\left(u+2\right)u^{2}}}$$

Do the multiplications in 6\left(u+2\right)+u^{2}u^{2}.

$$\frac{\frac{u^{2}+8u+4}{\left(u+2\right)u^{2}}}{\frac{6u+12+u^{4}}{\left(u+2\right)u^{2}}}$$

Divide \frac{u^{2}+8u+4}{\left(u+2\right)u^{2}} by \frac{6u+12+u^{4}}{\left(u+2\right)u^{2}} by multiplying \frac{u^{2}+8u+4}{\left(u+2\right)u^{2}} by the reciprocal of \frac{6u+12+u^{4}}{\left(u+2\right)u^{2}}.

$$\frac{\left(u^{2}+8u+4\right)\left(u+2\right)u^{2}}{\left(u+2\right)u^{2}\left(6u+12+u^{4}\right)}$$

Cancel out \left(u+2\right)u^{2} in both numerator and denominator.

$$\frac{u^{2}+8u+4}{u^{4}+6u+12}$$

Expand

$$\frac{u^{2}+8u+4}{u^{4}+6u+12}$$

Short Solution Steps

Factor u^{2}+2u.

$$\frac{\frac{u+2}{u^{2}}+\frac{4}{u\left(u+2\right)}}{\frac{6}{u^{2}}+\frac{u^{2}}{u+2}}$$

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

$$\frac{\frac{\left(u+2\right)\left(u+2\right)}{\left(u+2\right)u^{2}}+\frac{4u}{\left(u+2\right)u^{2}}}{\frac{6}{u^{2}}+\frac{u^{2}}{u+2}}$$

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

$$\frac{\frac{\left(u+2\right)\left(u+2\right)+4u}{\left(u+2\right)u^{2}}}{\frac{6}{u^{2}}+\frac{u^{2}}{u+2}}$$

Do the multiplications in \left(u+2\right)\left(u+2\right)+4u.

$$\frac{\frac{u^{2}+2u+2u+4+4u}{\left(u+2\right)u^{2}}}{\frac{6}{u^{2}}+\frac{u^{2}}{u+2}}$$

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

$$\frac{\frac{u^{2}+8u+4}{\left(u+2\right)u^{2}}}{\frac{6}{u^{2}}+\frac{u^{2}}{u+2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of u^{2} and u+2 is \left(u+2\right)u^{2}. Multiply \frac{6}{u^{2}} times \frac{u+2}{u+2}. Multiply \frac{u^{2}}{u+2} times \frac{u^{2}}{u^{2}}.

$$\frac{\frac{u^{2}+8u+4}{\left(u+2\right)u^{2}}}{\frac{6\left(u+2\right)}{\left(u+2\right)u^{2}}+\frac{u^{2}u^{2}}{\left(u+2\right)u^{2}}}$$

Since \frac{6\left(u+2\right)}{\left(u+2\right)u^{2}} and \frac{u^{2}u^{2}}{\left(u+2\right)u^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{u^{2}+8u+4}{\left(u+2\right)u^{2}}}{\frac{6\left(u+2\right)+u^{2}u^{2}}{\left(u+2\right)u^{2}}}$$

Do the multiplications in 6\left(u+2\right)+u^{2}u^{2}.

$$\frac{\frac{u^{2}+8u+4}{\left(u+2\right)u^{2}}}{\frac{6u+12+u^{4}}{\left(u+2\right)u^{2}}}$$

Divide \frac{u^{2}+8u+4}{\left(u+2\right)u^{2}} by \frac{6u+12+u^{4}}{\left(u+2\right)u^{2}} by multiplying \frac{u^{2}+8u+4}{\left(u+2\right)u^{2}} by the reciprocal of \frac{6u+12+u^{4}}{\left(u+2\right)u^{2}}.

$$\frac{\left(u^{2}+8u+4\right)\left(u+2\right)u^{2}}{\left(u+2\right)u^{2}\left(6u+12+u^{4}\right)}$$

Cancel out \left(u+2\right)u^{2} in both numerator and denominator.

$$\frac{u^{2}+8u+4}{u^{4}+6u+12}$$