Solve 2n/n+2n+m/2n-m+4mn/4n^2-n^2

$$\frac { 2 n } { n + 2 n } + \frac { m } { 2 n - m } + \frac { 4 m n } { 4 n ^ { 2 } - n ^ { 2 } }$$

Evaluate

$$\frac{4n^{2}+9mn-4m^{2}}{3n\left(2n-m\right)}$$

Short Solution Steps

Combine n and 2n to get 3n.

$$\frac{2n}{3n}+\frac{m}{2n-m}+\frac{4mn}{4n^{2}-n^{2}}$$

Cancel out n in both numerator and denominator.

$$\frac{2}{3}+\frac{m}{2n-m}+\frac{4mn}{4n^{2}-n^{2}}$$

Combine 4n^{2} and -n^{2} to get 3n^{2}.

$$\frac{2}{3}+\frac{m}{2n-m}+\frac{4mn}{3n^{2}}$$

Cancel out n in both numerator and denominator.

$$\frac{2}{3}+\frac{m}{2n-m}+\frac{4m}{3n}$$

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

$$\frac{2\left(-m+2n\right)}{3\left(-m+2n\right)}+\frac{3m}{3\left(-m+2n\right)}+\frac{4m}{3n}$$

Since \frac{2\left(-m+2n\right)}{3\left(-m+2n\right)} and \frac{3m}{3\left(-m+2n\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(-m+2n\right)+3m}{3\left(-m+2n\right)}+\frac{4m}{3n}$$

Do the multiplications in 2\left(-m+2n\right)+3m.

$$\frac{-2m+4n+3m}{3\left(-m+2n\right)}+\frac{4m}{3n}$$

Combine like terms in -2m+4n+3m.

$$\frac{m+4n}{3\left(-m+2n\right)}+\frac{4m}{3n}$$

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

$$\frac{\left(m+4n\right)n}{3n\left(-m+2n\right)}+\frac{4m\left(-m+2n\right)}{3n\left(-m+2n\right)}$$

Since \frac{\left(m+4n\right)n}{3n\left(-m+2n\right)} and \frac{4m\left(-m+2n\right)}{3n\left(-m+2n\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(m+4n\right)n+4m\left(-m+2n\right)}{3n\left(-m+2n\right)}$$

Do the multiplications in \left(m+4n\right)n+4m\left(-m+2n\right).

$$\frac{mn+4n^{2}-4m^{2}+8mn}{3n\left(-m+2n\right)}$$

Combine like terms in mn+4n^{2}-4m^{2}+8mn.

$$\frac{-4m^{2}+9mn+4n^{2}}{3n\left(-m+2n\right)}$$

Expand 3n\left(-m+2n\right).

$$\frac{-4m^{2}+9mn+4n^{2}}{-3mn+6n^{2}}$$

Differentiate w.r.t. m

$$\frac{2\left(-2m^{2}+8mn-11n^{2}\right)}{3n\left(m-2n\right)\left(2n-m\right)}$$